From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!
We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.

Logarithmic Functions

Logarithmic functions come in different flavors and bases. There's vanilla, strawberry...sugar cone, dish...but we'll start with the most basic one:

f(x) = ln x.

This function is defined for x > 0, and looks like this:

Think about the slope/derivative of this function. First off, since the function f(x) = ln x is always increasing, its derivative is always positive. Also, since f(x) = ln x is only defined for x > 0, f ' will also only be defined for x > 0. The graph of f ' will be entirely in the first quadrant:

If we take x = a close to zero, then the slope of f at a will be very steep:

Therefore f ' (a) will be large:

As x = a gets closer to zero, f ' (a) will be even larger:

If we take x = a far from zero, then the slope of f at a will be shallow, f ' (a) will be close to zero:

As x = a gets farther from zero, f ' (a) will move closer to zero:

If we fill in this rather sketchy graph of f ', we find the graph of  for x > 0:

Here's the rule for finding the derivative of the natural log function:

If f(x) = ln x, then  

The graph is useful for remembering this rule. After we introduce the chain rule we'll see another way to find the derivative of ln x.

People who Shmooped this also Shmooped...