list of logarithmic identities

Trivial mathematical identities are relatively simple (for an experienced mathematician), though not necessarily unimportant. The trivial logarithmic identities are as follows:

:

By definition, we know that:

:$\backslash log\_b(y)\; =\; x\; \backslash iff\; b^x\; =\; y$,

where $b\; \backslash neq\; 0$ and $b\; \backslash neq\; 1$.

Setting $x\; =\; 0$,

we can see that:

b^x = y

\iff b^{(0)} = y

\iff 1 = y

\iff y = 1

. So, substituting these values into the formula, we see that:

\log_b (y) = x

\iff \log_b (1) = 0

, which gets us the first property.

Setting $x\; =\; 1$,

we can see that:

b^x = y

\iff b^{(1)} = y

\iff b = y

\iff y = b

. So, substituting these values into the formula, we see that:

\log_b (y) = x

\iff \log_b (b) = 1

, which gets us the second property.

Logarithms and exponentials with the same base cancel each other. This is true because logarithms and exponentials are inverse operations—much like the same way multiplication and division are inverse operations, and addition and subtraction are inverse operations.

:$b^\{\backslash log\_b(x)\}\; =\; x\backslash text\{\; because\; \}\backslash mbox\{antilog\}\_b(\backslash log\_b(x))\; =\; x$

:$\backslash log\_b(b^x)\; =\; x\backslash text\{\; because\; \}\backslash log\_b(\backslash mbox\{antilog\}\_b(x))\; =\; x${{Cite web|last=Weisstein|first=Eric W.|title=Logarithm|url=https://mathworld.wolfram.com/Logarithm.html|access-date=2020-08-29|website=mathworld.wolfram.com|language=en}}

Both of the above are derived from the following two equations that define a logarithm:

(note that in this explanation, the variables of $x$ and $x$ may not be referring to the same number)

:$\backslash log\_b\; (y)\; =\; x\; \backslash iff\; b^x\; =\; y$

Looking at the equation $b^x\; =\; y$, and substituting the value for $x$ of

$\backslash log\_b\; (y)\; =\; x$, we get the following equation:

b^x = y

\iff b^{\log _b(y)} = y

\iff b^{\log_b (y)} = y

, which gets us the first equation.

Another more rough way to think about it is that $b^\{\backslash text\{something\}\}\; =\; y$,

and that that "$\backslash text\{something\}$" is $\backslash log\_b\; (y)$.

Looking at the equation $\backslash log\_b\; (y)\; =\; x$

, and substituting the value for $y$ of $b^x\; =\; y$, we get the following equation:

\log_b (y) = x

\iff \log_b(b^x) = x

\iff \log_b(b^x) = x

, which gets us the second equation.

Another more rough way to think about it is that $\backslash log\_b\; (\backslash text\{something\})\; =\; x$,

and that that something "$\backslash text\{something\}$" is $b^x$.

Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below.{{Cite web|title=4.3 - Properties of Logarithms|url=https://people.richland.edu/james/lecture/m116/logs/properties.html|access-date=2020-08-29|website=people.richland.edu}} The first three operations below assume that {{math|1=x = b^c}} and/or {{math|1=y = b^d}}, so that {{math|1=log_b(x) = c}} and {{math|1=log_b(y) = d}}. Derivations also use the log definitions {{math|1=x = b^log_b(x)}} and {{math|1=x = log_b(b^x)}}.

:

Where $b$, $x$, and $y$ are positive real numbers and $b\; \backslash ne\; 1$, and $c$ and $d$ are real numbers.

The laws result from canceling exponentials and the appropriate law of indices. Starting with the first law:

:$xy\; =\; b^\{\backslash log\_b(x)\}\; b^\{\backslash log\_b(y)\}\; =\; b^\{\backslash log\_b(x)\; +\; \backslash log\_b(y)\}\; \backslash Rightarrow\; \backslash log\_b(xy)\; =\; \backslash log\_b(b^\{\backslash log\_b(x)\; +\; \backslash log\_b(y)\})\; =\; \backslash log\_b(x)\; +\; \backslash log\_b(y)$

The law for powers exploits another of the laws of indices:

:$x^y\; =\; (b^\{\backslash log\_b(x)\})^y\; =\; b^\{y\; \backslash log\_b(x)\}\; \backslash Rightarrow\; \backslash log\_b(x^y)\; =\; y\; \backslash log\_b(x)$

The law relating to quotients then follows:

:$\backslash log\_b\; \backslash bigg(\backslash frac\{x\}\{y\}\backslash bigg)\; =\; \backslash log\_b(x\; y^\{-1\})\; =\; \backslash log\_b(x)\; +\; \backslash log\_b(y^\{-1\})\; =\; \backslash log\_b(x)\; -\; \backslash log\_b(y)$

:$\backslash log\_b\; \backslash bigg(\backslash frac\{1\}\{y\}\backslash bigg)\; =\; \backslash log\_b(y^\{-1\})\; =\; -\; \backslash log\_b(y)$

Similarly, the root law is derived by rewriting the root as a reciprocal power:

:$\backslash log\_b(\backslash sqrt[y]x)\; =\; \backslash log\_b(x^\{\backslash frac\{1\}\{y\}\})\; =\; \backslash frac\{1\}\{y\}\backslash log\_b(x)$

These are the three main logarithm laws/rules/principles,

{{Cite web

|url=https://courseware.cemc.uwaterloo.ca/8/assignments/210/6

|title=Properties and Laws of Logarithms

|website=courseware.cemc.uwaterloo.ca/8

|access-date=2022-04-23

}}

from which the other properties listed above can be proven. Each of these logarithm properties correspond to their respective exponent law, and their derivations/proofs will hinge on those facts. There are multiple ways to derive/prove each logarithm law – this is just one possible method.

To state the logarithm of a product law formally:

:$\backslash forall\; b\; \backslash in\; \backslash mathbb\{R\}\_+,\; b\; \backslash neq\; 1,\; \backslash forall\; x,\; y,\; \backslash in\; \backslash mathbb\{R\}\_+,\; \backslash log\_b(xy)\; =\; \backslash log\_b(x)\; +\; \backslash log\_b(y)$

Derivation:

Let $b\; \backslash in\; \backslash mathbb\{R\}\_+$, where $b\; \backslash neq\; 1$,

and let $x,\; y\; \backslash in\; \backslash mathbb\{R\}\_+$. We want to relate the expressions $\backslash log\_b(x)$ and $\backslash log\_b(y)$. This can be done more easily by rewriting in terms of exponentials, whose properties we already know. Additionally, since we are going to refer to $\backslash log\_b(x)$ and $\backslash log\_b(y)$ quite often, we will give them some variable names to make working with them easier: Let $m\; =\; \backslash log\_b(x)$, and let $n\; =\; \backslash log\_b(y)$.

Rewriting these as exponentials, we see that

:$\backslash begin\{align\}$

m &= \log_b(x) \iff b^m = x, \\

n &= \log_b(y) \iff b^n = y.

\end{align}

From here, we can relate $b^m$ (i.e. $x$) and $b^n$ (i.e. $y$) using exponent laws as

:$xy\; =\; (b^m)(b^n)\; =\; b^m\; \backslash cdot\; b^n\; =\; b^\{m\; +\; n\}$

To recover the logarithms, we apply $\backslash log\_b$ to both sides of the equality.

:$\backslash log\_b(xy)\; =\; \backslash log\_b(b^\{m\; +\; n\})$

The right side may be simplified using one of the logarithm properties from before: we know that $\backslash log\_b(b^\{m\; +\; n\})\; =\; m\; +\; n$, giving

:$\backslash log\_b(xy)\; =\; m\; +\; n$

We now resubstitute the values for $m$ and $n$ into our equation, so our final expression is only in terms of $x$, $y$, and $b$.

:$\backslash log\_b(xy)\; =\; \backslash log\_b(x)\; +\; \backslash log\_b(y)$

This completes the derivation.

To state the logarithm of a quotient law formally:

:$\backslash forall\; b\; \backslash in\; \backslash mathbb\{R\}\_+,\; b\; \backslash neq\; 1,\; \backslash forall\; x,\; y,\; \backslash in\; \backslash mathbb\{R\}\_+,\; \backslash log\_b\; \backslash left(\; \backslash frac\{x\}\{y\}\; \backslash right)\; =\; \backslash log\_b(x)\; -\; \backslash log\_b(y)$

Derivation:

Let $b\; \backslash in\; \backslash mathbb\{R\}\_+$, where $b\; \backslash neq\; 1$,

and let $x,\; y\; \backslash in\; \backslash mathbb\{R\}\_+$.

We want to relate the expressions $\backslash log\_b(x)$ and $\backslash log\_b(y)$. This can be done more easily by rewriting in terms of exponentials, whose properties we already know. Additionally, since we are going to refer to $\backslash log\_b(x)$ and $\backslash log\_b(y)$ quite often, we will give them some variable names to make working with them easier: Let $m\; =\; \backslash log\_b(x)$, and let $n\; =\; \backslash log\_b(y)$.

Rewriting these as exponentials, we see that:

:$\backslash begin\{align\}$

m &= \log_b(x) \iff b^m = x, \\

n &= \log_b(y) \iff b^n = y.

\end{align}

From here, we can relate $b^m$ (i.e. $x$) and $b^n$ (i.e. $y$) using exponent laws as

:$\backslash frac\{x\}\{y\}\; =\; \backslash frac\{(b^m)\}\{(b^n)\}\; =\; \backslash frac\{b^m\}\{b^n\}\; =\; b^\{m\; -\; n\}$

To recover the logarithms, we apply $\backslash log\_b$ to both sides of the equality.

:$\backslash log\_b\; \backslash left(\; \backslash frac\{x\}\{y\}\; \backslash right)\; =\; \backslash log\_b\; \backslash left(\; b^\{m\; -n\}\; \backslash right)$

The right side may be simplified using one of the logarithm properties from before: we know that $\backslash log\_b(b^\{m\; -\; n\})\; =\; m\; -\; n$, giving

:$\backslash log\_b\; \backslash left(\; \backslash frac\{x\}\{y\}\; \backslash right)\; =\; m\; -n$

We now resubstitute the values for $m$ and $n$ into our equation, so our final expression is only in terms of $x$, $y$, and $b$.

:$\backslash log\_b\; \backslash left(\; \backslash frac\{x\}\{y\}\; \backslash right)\; =\; \backslash log\_b(x)\; -\; \backslash log\_b(y)$

This completes the derivation.

To state the logarithm of a power law formally:

:$\backslash forall\; b\; \backslash in\; \backslash mathbb\{R\}\_+,\; b\; \backslash neq\; 1,\; \backslash forall\; x\; \backslash in\; \backslash mathbb\{R\}\_+,\; \backslash forall\; r\; \backslash in\; \backslash mathbb\{R\},\; \backslash log\_b(x^r)\; =\; r\backslash log\_b(x)$

Derivation:

Let $b\; \backslash in\; \backslash mathbb\{R\}\_+$, where $b\; \backslash neq\; 1$, let $x\backslash in\; \backslash mathbb\{R\}\_+$, and let $r\; \backslash in\; \backslash mathbb\{R\}$. For this derivation, we want to simplify the expression $\backslash log\_b(x^r)$. To do this, we begin with the simpler expression $\backslash log\_b(x)$. Since we will be using $\backslash log\_b(x)$ often, we will define it as a new variable: Let $m\; =\; \backslash log\_b(x)$.

To more easily manipulate the expression, we rewrite it as an exponential. By definition, $m\; =\; \backslash log\_b(x)\; \backslash iff\; b^m\; =\; x$, so we have

:$b^m\; =\; x$

Similar to the derivations above, we take advantage of another exponent law. In order to have $x^r$ in our final expression, we raise both sides of the equality to the power of $r$:

:$$

\begin{align}

(b^m)^r &= (x)^r \\

b^{mr} &= x^r

\end{align}

where we used the exponent law $(b^m)^r\; =\; b^\{mr\}$.

To recover the logarithms, we apply $\backslash log\_b$ to both sides of the equality.

:$\backslash log\_b(b^\{mr\})\; =\; \backslash log\_b(x^r)$

The left side of the equality can be simplified using a logarithm law, which states that $\backslash log\_b(b^\{mr\})\; =\; mr$.

:$mr\; =\; \backslash log\_b(x^r)$

Substituting in the original value for $m$, rearranging, and simplifying gives

:$$

\begin{align}

\left( \log_b(x) \right)r &= \log_b(x^r) \\

r\log_b(x) &= \log_b(x^r) \\

\log_b(x^r) &= r\log_b(x)

\end{align}

This completes the derivation.

To state the change of base logarithm formula formally:

$$\backslash forall\; a,\; b\; \backslash in\; \backslash mathbb\{R\}\_+,\; a,\; b\; \backslash neq\; 1,\; \backslash forall\; x\; \backslash in\; \backslash mathbb\{R\}\_+,\; \backslash log\_b(x)\; =\; \backslash frac\{\backslash log\_a(x)\}\{\backslash log\_a(b)\}$$

This identity is useful to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log₁₀, but not all calculators have buttons for the logarithm of an arbitrary base.

Let $a,\; b\; \backslash in\; \backslash mathbb\{R\}\_+$, where $a,\; b\; \backslash neq\; 1$ Let $x\; \backslash in\; \backslash mathbb\{R\}\_+$. Here, $a$ and $b$ are the two bases we will be using for the logarithms. They cannot be 1, because the logarithm function is not well defined for the base of 1.{{Citation needed|date=July 2022}} The number $x$ will be what the logarithm is evaluating, so it must be a positive number. Since we will be dealing with the term $\backslash log\_b(x)$ quite frequently, we define it as a new variable: Let $m\; =\; \backslash log\_b(x)$.

To more easily manipulate the expression, it can be rewritten as an exponential.

$$b^m\; =\; x$$

Applying $\backslash log\_a$ to both sides of the equality,

$$\backslash log\_a(b^m)\; =\; \backslash log\_a(x)$$

Now, using the logarithm of a power property, which states that $\backslash log\_a(b^m)\; =\; m\backslash log\_a(b)$,

$$m\backslash log\_a(b)\; =\; \backslash log\_a(x)$$

Isolating $m$, we get the following:

$$m\; =\; \backslash frac\{\backslash log\_a(x)\}\{\backslash log\_a(b)\}$$

Resubstituting $m\; =\; \backslash log\_b(x)$ back into the equation,

$$\backslash log\_b(x)\; =\; \backslash frac\{\backslash log\_a(x)\}\{\backslash log\_a(b)\}$$

This completes the proof that $\backslash log\_b(x)\; =\; \backslash frac\{\backslash log\_a(x)\}\{\backslash log\_a(b)\}$.

This formula has several consequences:

$$\backslash log\_b\; a\; =\; \backslash frac\; 1\; \{\backslash log\_a\; b\}$$

$$\backslash log\_\{b^n\}\; a\; =\; \{\backslash log\_b\; a\; \backslash over\; n\}$$

$$\backslash log\_\{b\}\; a\; =\; \backslash log\_b\; e\; \backslash cdot\; \backslash log\_e\; a\; =\; \backslash log\_b\; e\; \backslash cdot\; \backslash ln\; a$$

$$b^\{\backslash log\_a\; d\}\; =\; d^\{\backslash log\_a\; b\}$$

$$-\backslash log\_b\; a\; =\; \backslash log\_b\; \backslash left(\{1\; \backslash over\; a\}\backslash right)\; =\; \backslash log\_\{1/b\}\; a$$

$$\backslash log\_\{b\_1\}a\_1\; \backslash ,\backslash cdots\backslash ,\; \backslash log\_\{b\_n\}a\_n$$

= \log_{b_{\pi(1)}}a_1\, \cdots\, \log_{b_{\pi(n)}}a_n,

where $\backslash pi$ is any permutation of the subscripts {{math|1, ..., n}}. For example

$$\backslash log\_b\; w\backslash cdot\; \backslash log\_a\; x\; \backslash cdot\; \backslash log\_d\; c\; \backslash cdot\; \backslash log\_d\; z$$

= \log_d w \cdot \log_b x \cdot \log_a c \cdot \log_d z.

The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities:

Note that the subtraction identity is not defined if $a=c$, since the logarithm of zero is not defined. Also note that, when programming, $a$ and $c$ may have to be switched on the right hand side of the equations if $c\; \backslash gg\; a$ to avoid losing the "1 +" due to rounding errors. Many programming languages have a specific log1p(x) function that calculates $\backslash log\_e\; (1+x)$ without underflow (when $x$ is small).

More generally:

$$\backslash log\; \_b\; \backslash sum\_\{i=0\}^N\; a\_i\; =\; \backslash log\_b\; a\_0\; +\; \backslash log\_b\; \backslash left(\; 1+\backslash sum\_\{i=1\}^N\; \backslash frac\{a\_i\}\{a\_0\}\; \backslash right)\; =\; \backslash log\; \_b\; a\_0\; +\; \backslash log\_b\; \backslash left(\; 1+\backslash sum\_\{i=1\}^N\; b^\{\backslash left(\; \backslash log\_b\; a\_i\; -\; \backslash log\; \_b\; a\_0\; \backslash right)\}\; \backslash right)$$

A useful identity involving exponents:

$$x^\{\backslash frac\{\backslash log(\backslash log(x))\}\{\backslash log(x)\}\}\; =\; \backslash log(x)$$

or more universally:

$$x^\{\backslash frac\{\backslash log(a)\}\{\backslash log(x)\}\}\; =\; a$$

$$\backslash frac\{1\}\{\backslash frac\{1\}\{\backslash log\_x(a)\}\; +\; \backslash frac\{1\}\{\backslash log\_y(a)\}\}\; =\; \backslash log\_\{xy\}(a)$$

$$\backslash frac\{1\}\{\backslash frac\{1\}\{\backslash log\_x(a)\}-\backslash frac\{1\}\{\backslash log\_y(a)\}\}\; =\; \backslash log\_\{\backslash frac\{x\}\{y\}\}(a)$$

Based on,{{citation

| last = Topsøe | first = Flemming

| editor1-last = Cho | editor1-first = Yeol Je

| editor2-last = Kim | editor2-first = Jong Kyu

| editor3-last = Dragomir | editor3-first = Sever S.

| contribution = Some bounds for the logarithmic function

| contribution-url = https://rgmia.org/papers/v7n2/pade.pdf

| isbn = 978-1-59454-874-1

| location = New York

| mr = 2349596

| pages = 137–151

| publisher = Nova Science Publishers

| title = Inequality theory and applications, Vol. 4: Papers from the 8th International Conference on Nonlinear Functional Analysis and Applications held at Gyeongsang National University, Chinju and Kyungnam University, Masan, August 9–13, 2004

| year = 2007}}

:$\backslash frac\{x\}\{1+x\}\; \backslash leq\; \backslash ln(1+x)$

\leq \frac{x(6+x)}{6+4x}

\leq x \mbox{ for all } {-1} < x

:$\backslash begin\{align\}$

\frac{2x}{2+x}&\leq3-\sqrt{\frac{27}{3+2x}}\leq\frac{x}{\sqrt{1+x+x^2/12}} \\[4pt]

&\leq \ln(1+x)\leq \frac{x}{\sqrt{1+x}}\leq \frac{x}{2}\frac{2+x}{1+x} \\[4pt]

&\text{ for } 0 \le x \text{, reverse for } {-1} < x \le 0

\end{align}

All are accurate around $x=0$, but not for large numbers.

The following identity relates log semiring to the min-plus semiring.$$\backslash lim\_\{T\; \backslash rightarrow\; 0\}\; -T\backslash log(e^\{-\backslash frac\{s\}\{T\}\}\; +\; e^\{-\backslash frac\{t\}\{T\}\})\; =\; \backslash mathrm\{min\}\backslash \{s,t\backslash \}$$

:$\backslash lim\_\{x\backslash to\; 0^+\}\backslash log\_a(x)=-\backslash infty\backslash quad\; \backslash mbox\{if\; \}\; a\; >\; 1$

:

:$\backslash lim\_\{x\backslash to\backslash infty\}\backslash log\_a(x)=\backslash infty\backslash quad\; \backslash mbox\{if\; \}\; a\; >\; 1$

:

:$\backslash lim\_\{x\backslash to\; \backslash infty\}x^b\backslash log\_a(x)=\backslash infty\backslash quad\; \backslash mbox\{if\; \}\; b\; >\; 0$

:$\backslash lim\_\{x\backslash to\backslash infty\}\backslash frac\{\backslash log\_a(x)\}\{x^b\}=0\backslash quad\; \backslash mbox\{if\; \}\; b\; >\; 0$

The last limit is often summarized as "logarithms grow more slowly than any power or root of x".

:$\{d\; \backslash over\; dx\}\; \backslash ln\; x\; =\; \{1\; \backslash over\; x\; \},\; x\; >\; 0$

:$\{d\; \backslash over\; dx\}\; \backslash ln\; |x|\; =\; \{1\; \backslash over\; x\; \},\; x\; \backslash neq\; 0$

:$\{d\; \backslash over\; dx\}\; \backslash log\_a\; x\; =\; \{1\; \backslash over\; x\; \backslash ln\; a\},\; x\; >\; 0,\; a\; >\; 0,\; \backslash text\{\; and\; \}\; a\backslash neq\; 1$

:$\backslash ln\; x\; =\; \backslash int\_1^x\; \backslash frac\; \{1\}\{t\}\backslash \; dt$

To modify the limits of integration to run from $x$ to $1$, we change the order of integration, which changes the sign of the integral:

:$-\backslash int\_1^x\; \backslash frac\{1\}\{t\}\; \backslash ,\; dt\; =\; \backslash int\_x^1\; \backslash frac\{1\}\{t\}\; \backslash ,\; dt$

Therefore:

:$\backslash ln\; \backslash frac\{1\}\{x\}\; =\; \backslash int\_x^1\; \backslash frac\{1\}\{t\}\; \backslash ,\; dt$

:$\backslash ln(n\; +\; 1)\; =$

:$\backslash lim\_\{k\; \backslash to\; \backslash infty\}\; \backslash sum\_\{i=1\}^\{k\}\; \backslash frac\{1\}\{x\_i\}\; \backslash Delta\; x\; =$

:$\backslash lim\_\{k\; \backslash to\; \backslash infty\}\; \backslash sum\_\{i=1\}^\{k\}\; \backslash frac\{1\}\{1\; +\; \backslash frac\{i-1\}\{k\}n\}\; \backslash cdot\; \backslash frac\{n\}\{k\}\; =$

:$\backslash lim\_\{k\; \backslash to\; \backslash infty\}\; \backslash sum\_\{x=1\}^\{k\; \backslash cdot\; n\}\; \backslash frac\{1\}\{1\; +\; \backslash frac\{x\}\{k\}\}\; \backslash cdot\; \backslash frac\{1\}\{k\}\; =$

:$\backslash lim\_\{k\; \backslash to\; \backslash infty\}\; \backslash sum\_\{x=1\}^\{k\; \backslash cdot\; n\}\; \backslash frac\{1\}\{k\; +\; x\}\; =\; \backslash lim\_\{k\; \backslash to\; \backslash infty\}\; \backslash sum\_\{x=k+1\}^\{k\; \backslash cdot\; n\; +\; k\}\; \backslash frac\{1\}\{x\}\; =\; \backslash lim\_\{k\; \backslash to\; \backslash infty\}\; \backslash sum\_\{x=k+1\}^\{k\; (n\; +\; 1)\}\; \backslash frac\{1\}\{x\}$

for $\backslash textstyle\; \backslash Delta\; x\; =\; \backslash frac\{n\}\{k\}$ and $x\_\{i\}$ is a sample point in each interval.

The natural logarithm $\backslash ln(1\; +\; x)$ has a well-known Taylor series{{cite web

| last = Weisstein

| first = Eric W.

| title = Mercator Series

| website = MathWorld--A Wolfram Web Resource

| url = https://mathworld.wolfram.com/MercatorSeries.html

| access-date = 2024-04-24

}} expansion that converges for $x$ in the open-closed interval {{open-closed|−1, 1}}:

$$\backslash ln(1\; +\; x)\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{(-1)^\{n+1\}\; x^n\}\{n\}\; =\; x\; -\; \backslash frac\{x^2\}\{2\}\; +\; \backslash frac\{x^3\}\{3\}\; -\; \backslash frac\{x^4\}\{4\}\; +\; \backslash frac\{x^5\}\{5\}\; -\; \backslash frac\{x^6\}\{6\}\; +\; \backslash cdots.$$

Within this interval, for $x\; =\; 1$, the series is conditionally convergent, and for all other values, it is absolutely convergent. For $x\; >\; 1$ or $x\; \backslash leq\; -1$, the series does not converge to $\backslash ln(1\; +\; x)$. In these cases, different representationsTo extend the utility of the Mercator series beyond its conventional bounds one can calculate $\backslash ln(1\; +\; x)$ for $x\; =\; -\backslash frac\{n\}\{n+1\}$ and $n\; \backslash geq\; 0$ and then negate the result, $\backslash ln\backslash left(\backslash frac\{1\}\{n+1\}\backslash right)$, to derive $\backslash ln(n\; +\; 1)$. For example, setting $x\; =\; -\backslash frac\{1\}\{2\}$ yields $\backslash ln\; 2\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash frac\{1\}\{n\; 2^n\}$. or methods must be used to evaluate the logarithm.

It is not uncommon in advanced mathematics, particularly in analytic number theory and asymptotic analysis, to encounter expressions involving differences or ratios of harmonic numbers at scaled indices.{{cite book

| last1 = Flajolet

| first1 = Philippe

| last2 = Sedgewick

| first2 = Robert

| title = Analytic Combinatorics

| year = 2009

| publisher = Cambridge University Press

| isbn = 978-0521898065

| page = 389

}} See page 117, and VI.8 definition of shifted harmonic numbers on page 389

The identity involving the limiting difference between harmonic numbers at scaled indices and its relationship to the logarithmic function provides an intriguing example of how discrete sequences can asymptotically relate to continuous functions. This identity is expressed as{{cite arXiv

|last=Deveci

|first=Sinan

|title=On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function

|year=2022

|eprint=2211.10751

|class=math.NT

}} See Theorem 5.2. on pages 22 - 23

:$\backslash lim\_\{\{k\; \backslash to\; \backslash infty\}\}\; (H\_\{\{k(n+1)\}\}\; -\; H\_k)\; =\; \backslash ln(n+1)$

which characterizes the behavior of harmonic numbers as they grow large. This approximation (which precisely equals $\backslash ln(n+1)$ in the limit) reflects how summation over increasing segments of the harmonic series exhibits integral properties, giving insight into the interplay between discrete and continuous analysis. It also illustrates how understanding the behavior of sums and series at large scales can lead to insightful conclusions about their properties. Here $H\_k$ denotes the $k$-th harmonic number, defined as

:$H\_k\; =\; \backslash sum\_\{\{j=1\}\}^k\; \backslash frac\{1\}\{j\}$

The harmonic numbers are a fundamental sequence in number theory and analysis, known for their logarithmic growth. This result leverages the fact that the sum of the inverses of integers (i.e., harmonic numbers) can be closely approximated by the natural logarithm function, plus a constant, especially when extended over large intervals.{{cite book

| last1 = Graham

| first1 = Ronald L.

| last2 = Knuth

| first2 = Donald E.

| last3 = Patashnik

| first3 = Oren

| title = Concrete Mathematics: A Foundation for Computer Science

| year = 1994

| publisher = Addison-Wesley

| isbn = 0-201-55802-5

| page = 429

}}

{{cite web

| url = http://mathworld.wolfram.com/HarmonicNumber.html

| title = Harmonic Number

| publisher = Wolfram MathWorld

| access-date = 2024-04-24

}} See formula 13.

As $k$ tends towards infinity, the difference between the harmonic numbers $H\_\{k(n+1)\}$ and $H\_k$ converges to a non-zero value. This persistent non-zero difference, $\backslash ln(n+1)$, precludes the possibility of the harmonic series approaching a finite limit, thus providing a clear mathematical articulation of its divergence.{{cite report

| last1 = Kifowit

| first1 = Steven J.

| title = More Proofs of Divergence of the Harmonic Series

| publisher = Prairie State College

| year = 2019

| url = https://stevekifowit.com/pubs/harm2.pdf

| access-date = 2024-04-24

}} See Proofs 23 and 24 for details on the relationship between harmonic numbers and logarithmic functions.

{{cite journal

| last1 = Bell

| first1 = Jordan

| last2 = Blåsjö

| first2 = Viktor

| title = Pietro Mengoli's 1650 Proof That the Harmonic Series Diverges

| journal = Mathematics Magazine

| volume = 91

| issue = 5

| year = 2018

| pages = 341–347

| doi = 10.1080/0025570X.2018.1506656

| jstor = 48665556

| hdl = 1874/407528

| url = https://www.jstor.org/stable/48665556

| access-date = 2024-04-24

| hdl-access = free

}}

The technique of approximating sums by integrals (specifically using the integral test or by direct integral approximation) is fundamental in deriving such results. This specific identity can be a consequence of these approximations, considering:

:$\backslash sum\_\{\{j=k+1\}\}^\{k(n+1)\}\; \backslash frac\{1\}\{j\}\; \backslash approx\; \backslash int\_k^\{k(n+1)\}\; \backslash frac\{dx\}\{x\}$

The limit explores the growth of the harmonic numbers when indices are multiplied by a scaling factor and then differenced. It specifically captures the sum from $k+1$ to $k(n+1)$:

:$H\_\{\{k(n+1)\}\}\; -\; H\_k\; =\; \backslash sum\_\{\{j=k+1\}\}^\{k(n+1)\}\; \backslash frac\{1\}\{j\}$

This can be estimated using the integral test for convergence, or more directly by comparing it to the integral of $1/x$ from $k$ to $k(n+1)$:

:$\backslash lim\_\{\{k\; \backslash to\; \backslash infty\}\}\; \backslash sum\_\{\{j=k+1\}\}^\{k(n+1)\}\; \backslash frac\{1\}\{j\}\; =\; \backslash int\_k^\{k(n+1)\}\; \backslash frac\{dx\}\{x\}\; =\; \backslash ln(k(n+1))\; -\; \backslash ln(k)\; =\; \backslash ln\backslash left(\backslash frac\{k(n+1)\}\{k\}\backslash right)\; =\; \backslash ln(n+1)$

As the window's lower bound begins at $k+1$ and the upper bound extends to $k(n+1)$, both of which tend toward infinity as $k\; \backslash to\; \backslash infty$, the summation window encompasses an increasingly vast portion of the smallest possible terms of the harmonic series (those with astronomically large denominators), creating a discrete sum that stretches towards infinity, which mirrors how continuous integrals accumulate value across an infinitesimally fine partitioning of the domain. In the limit, the interval is effectively from $1$ to $n+1$ where the onset $k$ implies this minimally discrete region.

The harmonic number difference formula for $\backslash ln(m)$ is an extension of the classic, alternating identity of $\backslash ln(2)$:

:$\backslash ln(2)\; =\; \backslash lim\_\{k\; \backslash to\; \backslash infty\}\; \backslash sum\_\{n=1\}^\{k\}\; \backslash left(\; \backslash frac\{1\}\{2n-1\}\; -\; \backslash frac\{1\}\{2n\}\; \backslash right)$

which can be generalized as the double series over the residues of $m$:

:$\backslash ln(m)\; =\; \backslash sum\_\{x\; \backslash in\; \backslash langle\; m\; \backslash rangle\; \backslash cap\; \backslash mathbb\{N\}\}\; \backslash sum\_\{r\; \backslash in\; \backslash mathbb\{Z\}\_m\; \backslash cap\; \backslash mathbb\{N\}\}\; \backslash left(\; \backslash frac\{1\}\{x-r\}\; -\; \backslash frac\{1\}\{x\}\; \backslash right)\; =\; \backslash sum\_\{x\; \backslash in\; \backslash langle\; m\; \backslash rangle\; \backslash cap\; \backslash mathbb\{N\}\}\; \backslash sum\_\{r\; \backslash in\; \backslash mathbb\{Z\}\_m\; \backslash cap\; \backslash mathbb\{N\}\}\; \backslash frac\{r\}\{x(x-r)\}$

where $\backslash langle\; m\; \backslash rangle$ is the principle ideal generated by $m$. Subtracting $\backslash textstyle\; \backslash frac\{1\}\{x\}$ from each term $\backslash textstyle\; \backslash frac\{1\}\{x-r\}$ (i.e., balancing each term with the modulus) reduces the magnitude of each term's contribution, ensuring convergence by controlling the series' tendency toward divergence as $m$ increases. For example:

:$\backslash ln(4)\; =\; \backslash lim\_\{k\; \backslash to\; \backslash infty\}\; \backslash sum\_\{n=1\}^\{k\}\; \backslash left(\; \backslash frac\{1\}\{4n-3\}\; -\; \backslash frac\{1\}\{4n\}\; \backslash right)\; +\; \backslash left(\; \backslash frac\{1\}\{4n-2\}\; -\; \backslash frac\{1\}\{4n\}\; \backslash right)\; +\; \backslash left(\; \backslash frac\{1\}\{4n-1\}\; -\; \backslash frac\{1\}\{4n\}\; \backslash right)$

This method leverages the fine differences between closely related terms to stabilize the series. The sum over all residues $r\; \backslash in\; \backslash N$ ensures that adjustments are uniformly applied across all possible offsets within each block of $m$ terms. This uniform distribution of the "correction" across different intervals defined by $x-r$ functions similarly to telescoping over a very large sequence. It helps to flatten out the discrepancies that might otherwise lead to divergent behavior in a straightforward harmonic series. Note that the structure of the summands of this formula matches those of the interpolated harmonic number $H\_x$ when both the domain and range are negated (i.e., $-H\_\{-x\}$). However, the interpretation and roles of the variables differ.

A fundamental feature of the proof is the accumulation of the subtrahends $\backslash frac\{1\}\{x\}$ into a unit fraction, that is, $\backslash frac\{m\}\{x\}\; =\; \backslash frac\{1\}\{n\}$ for $m\; \backslash mid\; x$, thus $m\; =\; \backslash omega\; +\; 1$ rather than $m\; =\; |\backslash mathbb\{Z\}\_\{m\}\; \backslash cap\; \backslash mathbb\{N\}|$, where the extrema of $\backslash mathbb\{Z\}\_\{m\}\; \backslash cap\; \backslash mathbb\{N\}$ are $[0,\; \backslash omega]$ if $\backslash mathbb\{N\}\; =\; \backslash mathbb\{N\}\_\{0\}$ and $[1,\; \backslash omega]$ otherwise, with the minimum of $0$ being implicit in the latter case due to the structural requirements of the proof. Since the cardinality of $\backslash mathbb\{Z\}\_\{m\}\; \backslash cap\; \backslash mathbb\{N\}$ depends on the selection of one of two possible minima, the integral $\backslash textstyle\; \backslash int\; \backslash frac\{1\}\{t\}\; dt$, as a set-theoretic procedure, is a function of the maximum $\backslash omega$ (which remains consistent across both interpretations) plus $1$, not the cardinality (which is ambiguous{{Cite arXiv |eprint=1102.0418 |first=Peter |last=Harremoës |title=Is Zero a Natural Number? |year=2011|class=math.HO }} A synopsis on the nature of 0 which frames the choice of minimum as the dichotomy between ordinals and cardinals.{{Cite journal |last=Barton |first=N. |year=2020 |title=Absence perception and the philosophy of zero |journal=Synthese |volume=197 |issue=9 |pages=3823–3850 |doi=10.1007/s11229-019-02220-x |pmc=7437648 |pmid=32848285}} See section 3.1 due to varying definitions of the minimum). Whereas the harmonic number difference computes the integral in a global sliding window, the double series, in parallel, computes the sum in a local sliding window—a shifting $m$-tuple—over the harmonic series, advancing the window by $m$ positions to select the next $m$-tuple, and offsetting each element of each tuple by $\backslash frac\{1\}\{m\}$ relative to the window's absolute position. The sum $\backslash sum\_\{n=1\}^\{k\}\; \backslash sum\; \backslash frac\{1\}\{x\; -\; r\}$ corresponds to $H\_\{km\}$ which scales $H\_\{m\}$ without bound. The sum $\backslash sum\_\{n=1\}^\{k\}\; -\backslash frac\{1\}\{n\}$ corresponds to the prefix $H\_\{k\}$ trimmed from the series to establish the window's moving lower bound $k+1$, and $\backslash ln(m)$ is the limit of the sliding window (the scaled, truncatedThe $k+1$ shift is characteristic of the right Riemann sum employed to prevent the integral from degenerating into the harmonic series, thereby averting divergence. Here, $-\backslash frac\{1\}\{n\}$ functions analogously, serving to regulate the series. The successor operation $m\; =\; \backslash omega\; +\; 1$ signals the implicit inclusion of the modulus $m$ (the region omitted from $\backslash mathbb\{N\}\_\{1\}$). The importance of this, from an axiomatic perspective, becomes evident when the residues of $m$ are formulated as $e^\{\backslash ln(\backslash omega\; +\; 1)\}$, where $\backslash omega\; +\; 1$ is bootstrapped by $\backslash omega\; =\; 0$ to produce the residues of modulus $m\; =\; \backslash omega\; =\; \backslash omega\_\{0\}\; +\; 1\; =\; 1$. Consequently, $\backslash omega$ represents a limiting value in this context. series):

$$\backslash begin\{align\}$$

\sum_{n=1}^k \sum_{r=1}^{\omega} \left( \frac{1}{mn - r} - \frac{1}{mn} \right)

&= \sum_{n=1}^k \sum_{r=0}^{\omega} \left( \frac{1}{mn - r} - \frac{1}{mn} \right) \\

&= \sum_{n=1}^k \left( -\frac{1}{n} + \sum_{r=0}^{\omega} \frac{1}{mn - r} \right) \\

&= -H_k + \sum_{n=1}^k \sum_{r=0}^{\omega} \frac{1}{mn - r} \\

&= -H_k + \sum_{n=1}^k \sum_{r=0}^{\omega} \frac{1}{(n-1)m + m - r} \\

&= -H_k + \sum_{n=1}^k \sum_{j=1}^m \frac{1}{(n-1)m + j} \\

&= -H_k + \sum_{n=1}^k \left( H_{nm} - H_{m(n-1)} \right) \\

&= -H_k + H_{mk}

\end{align}

$$\backslash lim\_\{k\; \backslash to\; \backslash infty\}\; H\_\{km\}\; -\; H\_k\; =\; \backslash sum\_\{x\; \backslash in\; \backslash langle\; m\; \backslash rangle\; \backslash cap\; \backslash mathbb\{N\}\}\; \backslash sum\_\{r\; \backslash in\; \backslash mathbb\{Z\}\_m\; \backslash cap\; \backslash mathbb\{N\}\}\; \backslash left(\; \backslash frac\{1\}\{x-r\}\; -\; \backslash frac\{1\}\{x\}\; \backslash right)\; =\; \backslash ln(\backslash omega\; +\; 1)\; =\; \backslash ln(m)$$

:$\backslash int\; \backslash ln\; x\; \backslash ,\; dx\; =\; x\; \backslash ln\; x\; -\; x\; +\; C\; =\; x(\backslash ln\; x\; -\; 1)\; +\; C$

:$\backslash int\; \backslash log\_a\; x\; \backslash ,\; dx\; =\; x\; \backslash log\_a\; x\; -\; \backslash frac\{x\}\{\backslash ln\; a\}\; +\; C\; =\; \backslash frac\{x\; (\backslash ln\; x\; -\; 1)\}\{\backslash ln\; a\}\; +\; C$

To remember higher integrals, it is convenient to define

:$x^\{\backslash left\; [n\; \backslash right]\}\; =\; x^\{n\}(\backslash log(x)\; -\; H\_n)$

where $H\_n$ is the n^th harmonic number:

:$x^\{\backslash left\; [\; 0\; \backslash right\; ]\}\; =\; \backslash log\; x$

:$x^\{\backslash left\; [\; 1\; \backslash right\; ]\}\; =\; x\; \backslash log(x)\; -\; x$

:$x^\{\backslash left\; [\; 2\; \backslash right\; ]\}\; =\; x^2\; \backslash log(x)\; -\; \backslash begin\{matrix\}\; \backslash frac\{3\}\{2\}\; \backslash end\{matrix\}x^2$

:$x^\{\backslash left\; [\; 3\; \backslash right\; ]\}\; =\; x^3\; \backslash log(x)\; -\; \backslash begin\{matrix\}\; \backslash frac\{11\}\{6\}\; \backslash end\{matrix\}x^3$

Then

:$\backslash frac\{d\}\{dx\}\backslash ,\; x^\{\backslash left[\; n\; \backslash right]\}\; =\; nx^\{\backslash left[\; n-1\; \backslash right]\}$

:$\backslash int\; x^\{\backslash left[\; n\; \backslash right]\}\backslash ,dx\; =\; \backslash frac\{x^\{\backslash left\; [\; n+1\; \backslash right\; ]\}\}\{n+1\}\; +\; C$

The identities of logarithms can be used to approximate large numbers. Note that {{math|1=log_b(a) + log_b(c) = log_b(ac)}}, where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, {{math|2^32,582,657 −1}}. To get the base-10 logarithm, we would multiply 32,582,657 by {{math|log₁₀(2)}}, getting {{math|1=9,808,357.09543 = 9,808,357 + 0.09543}}. We can then get {{math|1=10^9,808,357 × 10^0.09543 ≈ 1.25 × 10^9,808,357}}.

Similarly, factorials can be approximated by summing the logarithms of the terms.

The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface. A single valued version, called the principal value of the logarithm, can be defined which is discontinuous on the negative x axis, and is equal to the multivalued version on a single branch cut.

In what follows, a capital first letter is used for the principal value of functions, and the lower case version is used for the multivalued function. The single valued version of definitions and identities is always given first, followed by a separate section for the multiple valued versions.

• {{math|ln(r)}} is the standard natural logarithm of the real number {{mvar|r}}.
• {{math|Arg(z)}} is the principal value of the arg function; its value is restricted to {{open-closed|−π, π}}. It can be computed using {{math|1=Arg(x + iy) = atan2(y, x)}}.
• {{math|Log(z)}} is the principal value of the complex logarithm function and has imaginary part in the range {{open-closed|−π, π}}.
• $\backslash operatorname\{Log\}(z)\; =\; \backslash ln(|z|)\; +\; i\; \backslash operatorname\{Arg\}(z)$
• $e^\{\backslash operatorname\{Log\}(z)\}\; =\; z$

The multiple valued version of {{math|log(z)}} is a set, but it is easier to write it without braces and using it in formulas follows obvious rules.

• {{math|log(z)}} is the set of complex numbers v which satisfy {{math|1=e^v = z}}
• {{math|arg(z)}} is the set of possible values of the arg function applied to z.

When k is any integer:

:$\backslash log(z)\; =\; \backslash ln(|z|)\; +\; i\; \backslash arg(z)$

:$\backslash log(z)\; =\; \backslash operatorname\{Log\}(z)\; +\; 2\; \backslash pi\; i\; k$

:$e^\{\backslash log(z)\}\; =\; z$

Principal value forms:

:$\backslash operatorname\{Log\}(1)\; =\; 0$

:$\backslash operatorname\{Log\}(e)\; =\; 1$

Multiple value forms, for any k an integer:

:$\backslash log(1)\; =\; 0\; +\; 2\; \backslash pi\; i\; k$

:$\backslash log(e)\; =\; 1\; +\; 2\; \backslash pi\; i\; k$

Principal value forms:

:$\backslash operatorname\{Log\}(z\_1)\; +\; \backslash operatorname\{Log\}(z\_2)\; =\; \backslash operatorname\{Log\}(z\_1\; z\_2)\; \backslash pmod\; \{2\; \backslash pi\; i\}$

:$\backslash operatorname\{Log\}(z\_1)\; +\; \backslash operatorname\{Log\}(z\_2)\; =\; \backslash operatorname\{Log\}(z\_1\; z\_2)\backslash quad$

(-\pi <\operatorname{Arg}(z_1)+\operatorname{Arg}(z_2)\leq \pi;

\text{ e.g., } \operatorname{Re}z_1\geq 0 \text{ and } \operatorname{Re}z_2 > 0){{Cite book|last=Abramowitz|first=Milton|url=https://www.worldcat.org/oclc/429082|title=Handbook of mathematical functions, with formulas, graphs, and mathematical tables|date=1965|publisher=Dover Publications|others=Irene A. Stegun|isbn=0-486-61272-4|location=New York|oclc=429082}}

:$\backslash operatorname\{Log\}(z\_1)\; -\; \backslash operatorname\{Log\}(z\_2)\; =\; \backslash operatorname\{Log\}(z\_1\; /\; z\_2)\; \backslash pmod\; \{2\; \backslash pi\; i\}$

:$\backslash operatorname\{Log\}(z\_1)\; -\; \backslash operatorname\{Log\}(z\_2)\; =\; \backslash operatorname\{Log\}(z\_1\; /\; z\_2)$

\quad

(-\pi <\operatorname{Arg}(z_1)-\operatorname{Arg}(z_2)\leq \pi;

\text{ e.g., } \operatorname{Re}z_1\geq 0 \text{ and } \operatorname{Re}z_2 > 0)

Multiple value forms:

:$\backslash log(z\_1)\; +\; \backslash log(z\_2)\; =\; \backslash log(z\_1\; z\_2)$

:$\backslash log(z\_1)\; -\; \backslash log(z\_2)\; =\; \backslash log(z\_1\; /\; z\_2)$

A complex power of a complex number can have many possible values.

Principal value form:

:$\{z\_1\}^\{z\_2\}\; =\; e^\{z\_2\; \backslash operatorname\{Log\}(z\_1)\}$

:$\backslash operatorname\{Log\}\{\backslash left(\{z\_1\}^\{z\_2\}\backslash right)\}\; =\; z\_2\; \backslash operatorname\{Log\}(z\_1)\; \backslash pmod\; \{2\; \backslash pi\; i\}$

Multiple value forms:

:$\{z\_1\}^\{z\_2\}\; =\; e^\{z\_2\; \backslash log(z\_1)\}$

Where {{math|k₁}}, {{math|k₂}} are any integers:

:$\backslash log\{\backslash left(\{z\_1\}^\{z\_2\}\backslash right)\}\; =\; z\_2\; \backslash log(z\_1)\; +\; 2\; \backslash pi\; i\; k\_2$

:$\backslash log\{\backslash left(\{z\_1\}^\{z\_2\}\backslash right)\}\; =\; z\_2\; \backslash operatorname\{Log\}(z\_1)\; +\; z\_2\; 2\; \backslash pi\; i\; k\_1\; +\; 2\; \backslash pi\; i\; k\_2$

As a consequence of the harmonic number difference, the natural logarithm is asymptotically approximated by a finite series difference, representing a truncation of the integral at $k\; =\; n$:

:$H\_\{\{2T[n]\}\}\; -\; H\_n\; \backslash sim\; \backslash ln(n+1)$

where $T[n]$ is the {{mvar|n}}th triangular number, and $2T[n]$ is the sum of the Pronic_number#As_figurate_numbers. Since the {{mvar|n}}th pronic number is asymptotically equivalent to the {{mvar|n}}th perfect square, it follows that:

:$H\_\{\{n^2\}\}\; -\; H\_n\; \backslash sim\; \backslash ln(n+1)$

The prime number theorem provides the following asymptotic equivalence:

:$\backslash frac\{n\}\{\backslash pi(n)\}\; \backslash sim\; \backslash ln\; n$

where $\backslash pi(n)$ is the prime counting function. This relationship is equal to:{{rp|2}}

:$\backslash frac\{n\}\{H(1,\; 2,\; \backslash ldots,\; x\_n)\}\; \backslash sim\; \backslash ln\; n$

where $H(x\_1,\; x\_2,\; \backslash ldots,\; x\_n)$ is the harmonic mean of $x\_1,\; x\_2,\; \backslash ldots,\; x\_n$. This is derived from the fact that the difference between the $n$th harmonic number and $\backslash ln\; n$ asymptotically approaches a small constant, resulting in $H\_\{\{n^2\}\}\; -\; H\_n\; \backslash sim\; H\_n$. This behavior can also be derived from the properties of logarithms: $\backslash ln\; n$ is half of $\backslash ln\; n^2$, and this "first half" is the natural log of the root of $n^2$, which corresponds roughly to the first $\backslash textstyle\; \backslash frac\{1\}\{n\}$th of the sum $H\_\{n^2\}$, or $H\_n$. The asymptotic equivalence of the first $\backslash textstyle\; \backslash frac\{1\}\{n\}$th of $H\_\{n^2\}$ to the latter $\backslash textstyle\; \backslash frac\{n-1\}\{n\}$th of the series is expressed as follows:

:$\backslash frac\{H\_n\}\{H\_\{n^2\}\}\; \backslash sim\; \backslash frac\{\backslash ln\; \backslash sqrt\{n\}\}\{\backslash ln\; n\}\; =\; \backslash frac\{1\}\{2\}$

which generalizes to:

:$\backslash frac\{H\_n\}\{H\_\{n^k\}\}\; \backslash sim\; \backslash frac\{\backslash ln\; \backslash sqrt[k]\{n\}\}\{\backslash ln\; n\}\; =\; \backslash frac\{1\}\{k\}$

:$k\; H\_n\; \backslash sim\; H\_\{n^k\}$

and:

:$k\; H\_n\; -\; H\_n\; \backslash sim\; (k\; -\; 1)\; \backslash ln(n+1)$

:$H\_\{\{n^k\}\}\; -\; H\_n\; \backslash sim\; (k\; -\; 1)\; \backslash ln(n+1)$

:$k\; H\_n\; -\; H\_\{\{n^k\}\}\; \backslash sim\; (k\; -\; 1)\; \backslash gamma$

for fixed $k$. The correspondence sets $H\_n$ as a unit magnitude that partitions $H\_\{n^k\}$ across powers, where each interval $\backslash textstyle\; \backslash frac\{1\}\{n\}$ to $\backslash textstyle\; \backslash frac\{1\}\{n^2\}$, $\backslash textstyle\; \backslash frac\{1\}\{n^2\}$ to $\backslash textstyle\; \backslash frac\{1\}\{n^3\}$, etc., corresponds to one $H\_n$ unit, illustrating that $H\_\{n^k\}$ forms a divergent series as $k\; \backslash to\; \backslash infty$.

These approximations extend to the real-valued domain through the interpolated harmonic number. For example, where $x\; \backslash in\; \backslash mathbb\{R\}$:

:$H\_\{\{x^2\}\}\; -\; H\_x\; \backslash sim\; \backslash ln\; x$

The natural logarithm is asymptotically related to the harmonic numbers by the Stirling numbers{{cite arXiv

| title = New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials

| author = Khristo N. Boyadzhiev

| year = 2022

| eprint = 1911.00186

| pages = 2, 6

| class = math.NT

}} and the Gregory coefficients.{{cite book

| last = Comtet

| first = Louis

| title = Advanced Combinatorics

| publisher = Kluwer

| year = 1974

}} By representing $H\_n$ in terms of Stirling numbers of the first kind, the harmonic number difference is alternatively expressed as follows, for fixed $k$:

:$\backslash frac\{s(n^k+1,\; 2)\}\{(n^k)!\}\; -\; \backslash frac\{s(n+1,\; 2)\}\{n!\}\; \backslash sim\; (k-1)\; \backslash ln(n+1)$

• {{annotated link|List of formulae involving π}}
• {{annotated link|List of integrals of logarithmic functions}}
• {{annotated link|List of mathematical identities}}
• {{annotated link|Lists of mathematics topics}}
• {{annotated link|List of trigonometric identities}}

{{reflist}}

• {{sister-inline

|project=v

|links=A lesson on logarithms can be found on Wikiversity

|short=yes

}}

• [http://www.mathwords.com/l/logarithm.htm Logarithm] in Mathwords

Category:Logarithms

Category:Mathematical identities

Category:Articles containing proofs