Logarithmic and exponential inverses $\newline$ Find the inverse of $\newline$ $f(x)=b^{x}$ and then prove that they are inverses.

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Q. Logarithmic and exponential inverses

\newline

Find the inverse of

\newline

f(x)=b^{x}

and then prove that they are inverses.

Understand Inverse Function Concept: Understand the concept of an inverse function. An inverse function reverses the effect of the original function. If $f(x) = y$ , then the inverse function $f^{-1}(y) = x$ . For the function $f(x) = b^x$ , we need to find a function that will give us the original $x$ when we input $y$ .
Express Function in Terms of y: Express the function in terms of y. $\newline$ Let's write the original function with y instead of f(x): $y = b^x$ .
Solve for x in Terms of y: Solve for x in terms of y. $\newline$ To find the inverse, we need to solve for x: $x = \log_b(y)$ . This means that the inverse function $f^{-1}(y)$ is the logarithm base $b$ of $y$ .
Write Inverse Function: Write the inverse function. $\newline$ The inverse function is $f^{-1}(y) = \log_b(y)$ .
Prove Functions are Inverses: Prove that $f$ and $f^{-1}$ are inverses. $\newline$ To prove that two functions are inverses, we need to show that $f(f^{-1}(y)) = y$ and $f^{-1}(f(x)) = x$ .
Apply $f$ to $f^{-1}$ : Apply $f$ to $f^{-1}$ . Let's apply $f$ to $f^{-1}(y)$ : $f(f^{-1}(y)) = f(\log_b(y)) = b^{\log_b(y)}$ .
Simplify Expression: Simplify the expression. $\newline$ Using the property of logarithms that $b^{\log_b(y)} = y$ , we simplify the expression to get $f(f^{-1}(y)) = y$ .
Apply $f^{-1}$ to $f$ : Apply $f^{-1}$ to $f$ . Now let's apply $f^{-1}$ to $f(x)$ : $f^{-1}(f(x)) = f^{-1}(b^x) = \log_b(b^x)$ .
Simplify Expression: Simplify the expression. Using the property of logarithms that $\log_b(b^x) = x$ , we simplify the expression to get $f^{-1}(f(x)) = x$ .