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.
#8^{\log_{8}(9)}=# #9#
This follows from the rule #a^{\log_a(x)}=x# for #a\gt0#, #a\neq1# and #x\gt0#. Take #a=8# and #b=9#.
This follows from the rule #a^{\log_a(x)}=x# for #a\gt0#, #a\neq1# and #x\gt0#. Take #a=8# and #b=9#.
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