# Logarithms and Exponential Functions

# Limits of Logarithmic Functions

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

Let's hold up the mirror by taking the base-4 logarithm to get *log _{x} = y*. If we tried to make

*x*negative or zero in either of these two equivalent functions, 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.

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

### Sample Problem

When will log_{5} *x* be greater than log_{2}0 *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 ; *5^{0}0.431or 20^{0}0.231

For *x* = 5; 5^{1} or 20^{0}0.537

However, when *x *is less than 1, log_{2}0 *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 *x* = ; 5^{-0.43} or 20^{-0.23}

For *x *= ; 5^{-0.86} or 20^{-0.46}