Finding derivatives of logs and natural logs

Formulas for the derivative of base-10 logs and natural logs

Given an exponential function in the form

$y=\log_{a}{\left[g(x)\right]}$

its derivative is

$y'=\left[\frac{1}{g(x)\ln{a}}\right]\left[g'(x)\right]$

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In the case that our base $a=e$, we have a special $\log$ that's called $\ln$. In other words, $\log_{e}{\left[g(x)\right]}=\ln{\left[g(x)\right]}$. The natural log function

$y=\ln{\left[g(x)\right]}$

has a derivative of

$y'=\left[\frac{1}{g(x)}\right]\left[g'(x)\right]$

Let’s try an example with $\log_a$.

Examples of how to find the derivative of different kinds of log and natural log functions

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Finding the derivative of a base-10 log function

Example

Find the derivative of the logarithmic function.

$y=6\log_8{\left(5x^4\right)}$

To find the derivative we need to apply the derivative formula for $\log$ functions.

$y'=6\left(\frac{1}{5x^4\ln{8}}\right)\left(20x^3\right)$

$y'=\frac{120x^3}{5x^4\ln{8}}$

$y'=\frac{24}{x\ln{8}}$

When the base of a log is e, instead of 10, we call it the “natural logarithm.”

Finding the derivative of a natural log function

Example

Find the derivative of the logarithmic function.

$y=5\ln{\left(2x^3\right)}$

To find the derivative we need to apply the derivative formula for natural logs.

$y'=5\left(\frac{1}{2x^3}\right)\left(6x^2\right)$

$y'=\frac{30x^2}{2x^3}$

$y'=\frac{15}{x}$

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Let’s try one final example that’s a little more complex.

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A more complicated logarithmic derivative

Example

Find the derivative of the logarithmic function.

$y=9\ln{\left(3x^7\right)}+\left(4x^7\right)\left[\log_3{\left(8x^2\right)}\right]-2x^{12}$

We need to take the derivative one term at a time, applying the derivative formulas from the beginning of this section.

$y'=9\left(\frac{1}{3x^7}\right)\left(21x^6\right)+\left[\left(28x^6\right)\left(\log_3{\left(8x^2\right)}\right)+\left(4x^7\right)\left(\frac{1}{8x^2\ln{3}}\right)(16x)\right]-24x^{11}$

$y'=\frac{189x^6}{3x^7}+28x^6\log_3{\left(8x^2\right)}+\frac{64x^8}{8x^2\ln{3}}-24x^{11}$

$y'=\frac{63}{x}+28x^6\log_3{\left(8x^2\right)}+\frac{8x^6}{\ln{3}}-24x^{11}$

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