# 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*.