Question

How do you find the inverse of \[y={{4}^{x}}\]?

Answer 1

Hint: Any function can have its inverse function, if it is a bijective function. A bijective function is both one-one and onto function. That is, each image of the function has distinct preimage and range equals to its co-domain. We can write the inverse function equation of an invertible function by expressing x in terms of y and then replacing with x.

Complete step by step answer:
As per the question, we are asked to find out the inverse function of the function \[y={{4}^{x}}\]. We know that the graph of \[y={{4}^{x}}\] function increases with x. So, we can say that it is an increasing function in its domain. As x value ranges from \[-\infty \] to \[\infty \], y value ranges from 0 to \[\infty \]. Hence, \[y={{4}^{x}}\] is an invertible function.
Now, let us find the inverse function of \[y={{4}^{x}}\].
Given equation is,
\[\Rightarrow y={{4}^{x}}\]
Here, we have 4 to the power x. So, we have to apply natural logarithm on both sides of the given equation. Then we get,
\[\Rightarrow \ln y=\ln {{4}^{x}}\] -------(1)
We know that, if we have the logarithm of a number ‘a’ to the power ‘b’, we have to write ‘b’ as the coefficient of the logarithm of ‘a’. That is, we can write
\[\Rightarrow \ln {{4}^{x}}=x\ln 4\] ------(2)
By substituting the equation (2) into the equation (1), we get
\[\Rightarrow \ln y=x\ln 4\]
\[\therefore x=\dfrac{\ln y}{\ln 4}\] --------(3)
We know that, \[\ln 4=2\ln 2=2\times 0.693\]. That is \[\ln 4=1.386\] .
\[\Rightarrow y=0.7213\ln x\]
By interchanging x and y we got the above equation.

\[\therefore y=0.7213\ln x\] is the required inverse function equation of \[y={{4}^{x}}\].

Note: We need to verify the inverse function obtained by composing \[f\] and \[{{f}^{-1}}\]. Common errors while composing functions: Students sometimes forget where each of the functions is defined before composing functions, which lead to non – existing results. They also sometimes forget that composition is not a commutative operation, that is, \[f\circ g\ne g\circ f\]. Also, the graphs of \[f\] and \[{{f}^{-1}}\] are symmetric about the line y=x.