Logarithmic Functions
We write the equation 
to say 
; that is, the expression 
asks, “a is equal to b to what power?” For example, 
and 
. As we’ve written it, b is called the base of the logarithm, and a is called the argument.
Observe that 
and 
; further, the domain of 
is the same as the range of 
and the domain of 
is the same as the range of 
.
In fact, a logarithmic function is the inverse function of an exponential function with the same base.
An exponential equation 
(naively) says, “Multiply a times itself x times to get y” while the logarithmic function 
says “To get x, multiply a by itself y times.”
Just as in the exponential case, we only use positive bases for logarithms. Indeed, since 
, we have 
.
Since 
is the inverse function of a smooth continuous function, it is a smooth continuous function. However, note that the domain of 
is all positive real numbers.
