# Limits of Logarithmic Functions

Just like exponential functions, **logarithmic functions** have their own limits. Remember what exponential functions can't do: they can't output a negative number for *f *(*x*). The function we took a gander at when thinking about exponential functions was *f *(*x*) = 4* ^{x}*.

Let's hold up the mirror by taking the base-4 logarithm to get the inverse function: *f *(*x*) = log_{4} *x*.

If we tried to make *x* negative or zero in this log function, there is no *y*-value in the known universe that would let us do it—so the log function is undefined at *x*-values of zero or less. In other words, its domain is *x* > 0.

Here's what the graph of *f *(*x*) = log_{4} *x* looks like:

Because the *output* of an exponential function can never be zero or negative, the inverse (log) function can never have a negative *input* of zero.

### Sample Problem

When will *f *(*x*) = log_{5} *x* be greater than *g *(*x*) = log_{20} *x*? Ignore negative *x*-values.

The output of these logs is the *exponent *needed above 5 or 20 to equal *x.*

When *x* is greater than 1, log_{5} *x *will be larger because a *larger exponent* is needed for 5 than for 20 to equal whatever *x *is in this domain. The 5 is a little shrimp that needs a boost, basically. For example:

For* x* = 2, we get 5^{0.43} or 20^{0.23}.

For *x* = 5, we get 5^{1} or 20^{0.54}.

However, when *x *is less than 1, log_{20} *x* will be greater than log_{5} *x* because a negative exponent closer to zero is needed to pull that 20 down to size versus the 5. That 20 has a lot of muscle, and needs a big push by the exponent to get down to size.

For , we get 5^{-0.43} or 20^{-0.23}.

For , we get 5^{-0.86} or 20^{-0.46}.