Given function:
[tex]f(x)=2^x[/tex]
To obtain the inverse of the function f(x), we follow the steps outlined below:
Step 1: Replace f(x) with y:
[tex]y=2^x[/tex]
Step 2: Interchange x and y
[tex]x=2^y[/tex]
Step 3: Solve for y:
[tex]\begin{gathered} \text{Take logarithm of both sides} \\ \log x=log2^y \\ y\log 2\text{ = log x} \\ \text{Divide both sides by log2} \\ y\text{ = }\frac{\log x}{\log \text{ 2}} \\ y\text{ = }\log _2x \end{gathered}[/tex]
Step 4: Replace y with f-1(x):
[tex]f^{-1}(x)\text{ = }\log _2x[/tex]
Answer:
[tex]f^{-1}(x)\text{ = }\log _2x[/tex]
