Question

How do you invert a logarithmic function?

Answer 1

Hint : To invert a logarithmic function, the main step is to isolate the log expression and then rewrite the log equation into an exponential equation. We can also check, whether the inverse of a function we have found is correct or not by making a graph of the log equation and inverse function in a single xy-axis and if the graphs formed are symmetrical to each other then we get assure that the inverse of the function we have found is correct.

Complete step by step solution:
The process to invert a logarithmic function is to replace the notation of the function $f\left( x \right)$ with y, which can also be written as $f\left( x \right) \to y$ , then we have to switch the roles of x and y, which is also written as,
$x \to y \\\ y \to x \;$
Now, we have to deal with the log expression separately on one side of the equation and then we need to convert the log equation into an exponential equation. If the log equation is ${\log _b}\left( M \right) = N$ then the exponential equation will be $M = {b^N}$ . Now, we have to solve the exponential equation for y to get the inverse of logarithmic function and after solving we have to replace $y$ by ${f^{ - 1}}\left( x \right)$ , here ${f^{ - 1}}\left( x \right)$ is the inverse notation for the final answer.

Note : In the problem, when we are converting the log equation into an exponential equation then the subscript $b$ of log has become the base with the exponent $N$ in the exponential form and the variable $M$ stays at the same place while converting. If in a logarithmic function the base of the log expression is missing then we can assume the base as $10$ .