Solving Exponential and Logarithmic Equations

By definition, logarithms are the inverse of exponents. Therefore, if we make an exponential function, f(x)=b^x, and its inverse logarithmic function, f^{-1}(x)=log_{b}(x), then…

    \[[f \circ f^{-1}](x)= b^{log_{b}(x)}=x\]

and

    \[[f^{-1} \circ f](x)= log_{b}(b^x)=x\]

Proof

    \[b^{log_{b}(x)}=x\]

if log_{b}(x)=y

\Rightarrow b^y=x

Since y=log_{b}(x)

\Rightarrow b^{log_{b}(x)}=b^y=x

    \[log_{b}(b^{x})=x\]

if b^{x}=y

\Rightarrow log_{b}(y)=x

since y=b^x

\Rightarrow log_{b}(b^{x})=log_{b}(y)=x