### Operations for functions: Exponential and logarithmic functions

### Logarithmic functions

If #a# is a positive number distinct from #1#, then, by the *properties of the exponents*, the *exponential function* #a^x# is *monotonic*, hence *injective* and, according to the *Characterization of invertible functions*, *invertible*.

Logarithmic function

Let #a# be a positive number distinct from #1#. The inverse function of #a^x# is called the **logarithmic function** to the **base** #a#, and denoted by #\log_a(x)#.

Thus, \[\begin{array}{rcl} a^{\log_a(y)}&=&y\phantom{xx}\text{ for all }y\gt 0\\ &\text{and}&\\ \log_a\left(a^x\right)&=&x\phantom{xx}\text{ for all real numbers }x\end{array}\]

The equalities are direct consequences from the definition of the *inverse of a function*.

Simplify # 9^{\log_{9}(5)}# to an integer.

Solution #9^{\log_{9}(5)}=# #5#

This follows from the rule #a^{\log_a(x)}=x# for #a\gt0#, #a\neq1# and #x\gt0#. Take #a=9# and #b=5#.

This follows from the rule #a^{\log_a(x)}=x# for #a\gt0#, #a\neq1# and #x\gt0#. Take #a=9# and #b=5#.

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