Logarithmic Functions

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We develop the rules needed to differentiate logarithmic functions.

Video Lecture

In this video we will develop a rule for taking a derivative of logs in general, starting with the natural log.

To see why implicit differentiation is an obvious tool to use when determining the derivative rules for logarithms, you can watch a supplemental video here:

https://youtu.be/YpZHKiNyrJ8

(Supplemental Videos are included via external link so you don’t have to watch them to earn credit.)

Text and Additional Details

We want to determine a rule for taking the derivative of $y = \ln (x)$ . Normally we would start with the difference quotient, and that approach is possible. But there is an easier way using a technique we’ve already developed - implicit differentiation. To this end, we will consider the exponential form of our equality: specifically, if $y = \ln (x)$ , then equivalently we can write that $e^y = x$ . We’ll use implicit differentiation to take the derivative of this equivalent form.

$\displaystyle \frac {d}{dx} [y] = \frac {d}{dx}\left [\ln (x)\right ]$
$\displaystyle \frac {d}{dx} \left [ e^y \right ] = \frac {d}{dx} [x]$	Exponential Form
$\displaystyle e^y\frac {d}{dx} \left [ y \right ] = 1$	Apply derivative and chain rule.
$\displaystyle e^y \cdot y' = 1$	Implicit Derivative.
$\displaystyle y' = \frac {1}{e^y}$	Divide both sides by $e^y$ .
$\displaystyle y' = \frac {1}{x}$	Recall exponential form $e^y = x$ for substitution.

Thus using implicit differentiation we have shown that the derivative of the natural log is $\frac {1}{x}$ , i.e.,

$\frac {d}{dx} \ln (x) = \frac {1}{x}$

In order to expand this to logs of any base, we only need to recall the change of base formula: $\log _b(x) = \frac {\ln (x)}{\ln (b)}$ . Using this we have the following:

$\displaystyle \frac {d}{dx} \left [\log _b(x)\right ]$
	$\displaystyle = \frac {d}{dx} \left [\frac {\ln (x)}{\ln (b)}\right ]$	Change of base
	$\displaystyle = \frac {d}{dx} \left [\frac {1}{\ln (b)}\ln (x)\right ]$	Broke up Fraction
	$\displaystyle = \frac {1}{\ln (b)} \cdot \frac {d}{dx} \left [\ln (x)\right ]$	Constant Multiple Rule
	$\displaystyle = \frac {1}{\ln (b)} \cdot \frac {1}{x}$	Derivative of natural log.

So now we can record our general formula for derivatives of logs:

Derivative of Logarithms Let $f(x) = \log _b(x)$ . Then

$\frac {d}{dx} f(x) = \frac {d}{dx} \left [\log _b(x)\right ] = \frac {1}{\ln (b)} \cdot \frac {1}{x}$ Moreover, in the special case where the base is $e$ and thus $f(x)$ is the natural log of $x$ , then we have:

$\frac {d}{dx} f(x) = \frac {d}{dx} \left [\ln (x)\right ] = \frac {1}{\ln (e)} \cdot \frac {1}{x} = \frac {1}{x}$

So, we have seen, that using implicit differentiation allows us to quickly determine the derivative of the natural log; and using that as a starting point, we were able to determine a rule for logarithms of any base!

1 : True or false, there are two different formulas for logs, one for taking the derivative of natural log, and a different formula for taking a derivative of a log with any other base.

There is only one formula. Notice that if you apply the general formula to the natural log (log with base $e$ ) then it still gives you the correct answer. However, given the frequency that natural logs are used in the real world, it often gets a special mention just because of how easy and elegant it is - and how often it ends up being used.

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