Notes

Questions

Find the inverse of the following logarithmic functions.

\(\textbf{1)}\) \(\hspace{1ex} f(x)=log(x) \)

Show Inverse

The answer is \( f^{-1}(x)=10^x \)

Show Work

\(\,\,\,\,\,\,f(x)=log(x) \)
\(\text{Step 1: Substitute } y \text{ for } f(x)\)
\(\,\,\,\,\,\,y=log(x) \)
\(\text{Step 2: Switch x and y.}\)
\(\,\,\,\,\,\,x=log(y)\)
\(\,\,\,\,\,\,x=log_{10}(y)\)
\(\text{Step 3: Use } log_b{a}=c \,\,\,\Rightarrow\,\,\, a=b^c\)
\(\,\,\,\,\,\,y=10^x\)
\(\text{Step 4: Rewrite } y \text{ as } f^{-1}(x)\)
\(\,\,\,\,\,\,f^{-1}(x)=10^x\)
The answer is \( f^{-1}(x)=10^x \)

 

\(\textbf{2)}\) \(\hspace{1ex} f(x)=log_{2}(x) \)

Show Inverse

The answer is \( f^{-1}(x)=2^x \)

 

\(\textbf{3)}\) \(\hspace{1ex} f(x)=log_{2}(x+2)+1 \)

Show Inverse

The answer is \( f^{-1}(x)=2^{x-1}-2 \)

 

\(\textbf{4)}\) \(\hspace{1ex} f(x)=2log_{2}(x-1)-3 \)

Show Inverse

The answer is \( f^{-1}(x)=2^{(\frac{x+3}{2})}+1 \)

 

\(\textbf{5)}\) \(\hspace{1ex} f(x)=ln(x) \)

Show Inverse

The answer is \( f^{-1}(x)=e^{x} \)

 

\(\textbf{6)}\) \(\hspace{1ex} f(x)=-ln(x) \)

Show Inverse

The answer is \( f^{-1}(x)=e^{-x} \)

 

\(\textbf{7)}\) \(\hspace{1ex} f(x)=ln(-x) \)

Show Inverse

The answer is \( f^{-1}(x)=-e^{x} \)

 

\(\textbf{8)}\) \(\hspace{1ex} f(x)=-ln(-x) \)

Show Inverse

The answer is \( f^{-1}(x)=-e^{-x} \)

 

 

See Related Pages\(\)

\(\bullet\text{ Logarithmic Form & Exponential Form}\)
\(\,\,\,\,\,\,\,\,\log_{b}(a)=c \rightarrow b^c=a…\)

\(\bullet\text{ Evaluating Logarithms}\)
\(\,\,\,\,\,\,\,\,\log_{2}(8)…\)

\(\bullet\text{ Expanding Logarithms}\)
\(\,\,\,\,\,\,\,\,2\log_{b}(x)+\log_{b}(z)-5\log_{b}(y)…\)

\(\bullet\text{ Decibel Problems}\)
\(\,\,\,\,\,\,\,\,N_{dB}=10\log \left(\frac{P}{10^{-12}}\right)…\)

\(\bullet\text{ Earthquake Problems}\)
\(\,\,\,\,\,\,\,\,M=\log\frac{I}{10^{-4}}…\)

\(\bullet\text{ Domain and Range Logarithmic Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=log(x) \rightarrow \text{Domain:} x\gt0… \)

\(\bullet\text{ Graphing Logarithmic Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=log_{2}(x)\)

\(\bullet\text{ Solving Logarithmic Equations}\)
\(\,\,\,\,\,\,\,\,\log_{2}(5x)=\log_{2}(2x+12)…\)

\(\bullet\text{ Inverse of Logarithmic Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=log_{2}(x) \rightarrow f^{-1}(x)=2^x\)