Notes
Questions
Find the inverse of the following logarithmic functions.
\(\textbf{1)}\) \(\hspace{1ex} f(x)=log(x) \)
The answer is \( f^{-1}(x)=10^x \) \(\,\,\,\,\,\,f(x)=log(x) \)
\(\text{Step 1: Substitute } y \text{ for } f(x)\)
\(\,\,\,\,\,\,y=log(x) \)
\(\text{Step 2: Switch x and y.}\)
\(\,\,\,\,\,\,x=log(y)\)
\(\,\,\,\,\,\,x=log_{10}(y)\)
\(\text{Step 3: Use } log_b{a}=c \,\,\,\Rightarrow\,\,\, a=b^c\)
\(\,\,\,\,\,\,y=10^x\)
\(\text{Step 4: Rewrite } y \text{ as } f^{-1}(x)\)
\(\,\,\,\,\,\,f^{-1}(x)=10^x\)
The answer is \( f^{-1}(x)=10^x \)
\(\text{Step 1: Substitute } y \text{ for } f(x)\)
\(\,\,\,\,\,\,y=log(x) \)
\(\text{Step 2: Switch x and y.}\)
\(\,\,\,\,\,\,x=log(y)\)
\(\,\,\,\,\,\,x=log_{10}(y)\)
\(\text{Step 3: Use } log_b{a}=c \,\,\,\Rightarrow\,\,\, a=b^c\)
\(\,\,\,\,\,\,y=10^x\)
\(\text{Step 4: Rewrite } y \text{ as } f^{-1}(x)\)
\(\,\,\,\,\,\,f^{-1}(x)=10^x\)
The answer is \( f^{-1}(x)=10^x \)
\(\textbf{2)}\) \(\hspace{1ex} f(x)=log_{2}(x) \)
The answer is \( f^{-1}(x)=2^x \)
\(\textbf{3)}\) \(\hspace{1ex} f(x)=log_{2}(x+2)+1 \)
The answer is \( f^{-1}(x)=2^{x-1}-2 \)
\(\textbf{4)}\) \(\hspace{1ex} f(x)=2log_{2}(x-1)-3 \)
The answer is \( f^{-1}(x)=2^{(\frac{x+3}{2})}+1 \)
\(\textbf{5)}\) \(\hspace{1ex} f(x)=ln(x) \)
The answer is \( f^{-1}(x)=e^{x} \)
\(\textbf{6)}\) \(\hspace{1ex} f(x)=-ln(x) \)
The answer is \( f^{-1}(x)=e^{-x} \)
\(\textbf{7)}\) \(\hspace{1ex} f(x)=ln(-x) \)
The answer is \( f^{-1}(x)=-e^{x} \)
\(\textbf{8)}\) \(\hspace{1ex} f(x)=-ln(-x) \)
The answer is \( f^{-1}(x)=-e^{-x} \)
