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