Logarithm rule, base a · Derivative Calculator

What it says

The change-of-base identity $\log_{a}x = \frac{\ln x}{\ln a}$ turns every logarithm into a constant multiple of $\ln x$. Since $\ln a$ is just a number, the constant multiple rule finishes the job: $\frac{1}{\ln a}\cdot\frac{1}{x}$.

So the only difference from the natural logarithm is a fixed divisor. For base $e$ it disappears, because $\ln e = 1$; for base $10$ it is $\ln 10 \approx 2.3026$, which is why $\log_{10}$ curves flatten so much faster.

In this calculator, $\log(a, x)$ means $\log_{a}x$, and $\log_{2}$ and $\log_{10}$ have their own shorthands $\texttt{log2}$ and $\texttt{log10}$.

When it applies

A logarithm whose base is a constant other than $e$.
$\log_{2}$ in computing work, $\log_{10}$ in decibels, pH and orders of magnitude.
If the base itself contains the variable, this rule does not apply — the expression is rewritten in natural logs instead.

Five worked examples

Every line is the step the calculator would show, in the order it applies them. Each graph is live: hover it to read both curves and see the tangent whose slope is the derivative, drag to pan, scroll to zoom.

Example 1

\frac{d}{dx}\left[\log_{2}\left(x\right)\right]

Change of base.
$\log_{2}x = \frac{\ln x}{\ln 2}$
$\frac{1}{\ln 2}$ is a constant factor.
$= \frac{1}{\ln 2}\cdot\frac{1}{x} = \frac{1}{x\ln 2}$

Answer

\frac{1}{x \ln\left(2\right)}

The shape of $\ln x$ stretched vertically by $\tfrac{1}{\ln 2} \approx 1.44$, with f′ correspondingly above the curve $\tfrac{1}{x}$. Changing the base rescales the whole picture; it does not bend it.

About $1.4427/x$ — steeper than $\ln x$, because $\log_{2}$ climbs faster.

Open log(2,x) in the calculator

Example 2

\frac{d}{dx}\left[\log_{10}\left(x\right)\right]

Same route, base $10$.
$= \frac{1}{x\ln 10}$

Answer

\frac{1}{x \ln\left(10\right)}

The flattest member of the family, and its derivative — about $\tfrac{0.434}{x}$ — is the smallest of them.

$\ln 10 \approx 2.3026$, so the common logarithm has under half the slope of $\ln x$.

Open log(10,x) in the calculator

Example 3

\frac{d}{dx}\left[\log_{2}\left(x^{3}\right)\right]

Use the power law first: $\log_{2}x^{3} = 3\log_{2}x$.
$= 3\;\frac{d}{dx}\left[\log_{2}x\right]$
Then the base-$a$ rule.
$= \frac{3}{x\ln 2}$

Answer

\frac{3}{x \ln\left(2\right)}

Three times the height of $\log_{2}x$ and three times the slope at every point, because $\log_{2}x^{3} = 3\log_{2}x$.

The chain rule gives the same thing: $\frac{3x^{2}}{x^{3}\ln 2} = \frac{3}{x\ln 2}$.

Open log(2,x^3) in the calculator

Example 4

\frac{d}{dx}\left[\log_{5}\left(3 x\right)\right]

Chain rule with $u = 3x$.
$= \frac{1}{3x\ln 5}\cdot 3$
The $3$ cancels.
$= \frac{1}{x\ln 5}$

Answer

\frac{1}{x \ln\left(5\right)}

Only for $x > 0$. The curve rises steeply and then settles into a near-straight climb, while f′ decays toward the axis.

As with $\ln$, a constant factor inside the logarithm shifts the graph but does not tilt it.

Open log(5,3x) in the calculator

Example 5

\frac{d}{dx}\left[x \log_{2}\left(x\right)\right]

Product rule with $f = x$, $g = \log_{2}x$.
$= 1 \cdot \log_{2}x + x\cdot\frac{1}{x\ln 2}$
The $x$ cancels in the second term.
$= \log_{2}x + \frac{1}{\ln 2}$

Answer

\log_{2}\left(x\right) + \frac{1}{\ln\left(2\right)}

The $n\log n$ curve familiar from sorting costs, turning at $x = \tfrac{1}{e}$ where f′ crosses zero.

The shape of $n\log_{2}n$, the cost of a merge sort — its derivative is what makes the curve just steeper than linear.

Open x log(2,x) in the calculator

Watch out for

The divisor is $\ln a$, the natural log of the base — not $\log a$ and not $a$.
$\log(a, x)$ here means base $a$; a single-argument $\log(x)$ means the natural logarithm.
A variable base needs a rewrite, not this rule.

Logarithm rule Rewrite in natural logarithms Constant multiple rule

What it says

When it applies

Five worked examples

Watch out for

Related rules