Question

What is the inverse of the function $y={{\log }_{4}}x$?

Answer 1

From the question we have been asked to find the inverse of the given logarithmic function. For solving the question we will use the logarithmic laws and we will interchange the variables so that it will be easier for us to solve the question. After interchanging by using the logarithmic laws we will simplify the equation and solve the given question.

Complete step-by-step solution:
We have that,
$\Rightarrow y={{\log }_{4}}x$
Now we will interchange the variables, that is we write $x$ in place of $y$ and we write $y$ in the place of $x$. So, we get the equation reduced after interchanging as follows.
$\Rightarrow x={{\log }_{4}}y$
Generally in mathematics for finding the inverse of a given function we will have to find the value of $y$ or simply we should solve for $y$. So, we get,
$\Rightarrow x={{\log }_{4}}y$
From the laws of the logarithm we have that for a logarithm of form $\Rightarrow x={{\log }_{z}}y$ we can remove the log and we can rewrite the function as $\Rightarrow {{z}^{x}}=y$.
So, now we will use this law and rewrite the above equation by removing the log. So, we get the equation reduced as follows.
$\Rightarrow y={{4}^{x}}$
So, now we will replace the $y$ with ${{f}^{-1}}\left( x \right)$
Therefore, we got the inverse function as ${{f}^{-1}}\left( x \right)={{4}^{x}}$.

Note: Students must be very careful in doing the calculations. Students must have good knowledge in the concept of logarithms. Students should know the laws of logarithm like we should know that, for a logarithm of form $x={{\log }_{z}}y$ we can remove the log and we can rewrite the function as ${{z}^{x}}=y$..