Question

How do you find the inverse of \[3^{2x}\] ?

Answer 1

Hint: Here in this question we have to find the inverse of the function, since the function is of the form exponential number we use the concept of logarithm function and on further simplification we obtain the required solution for the given question. To find the inverse we swap the terms.

Complete step by step solution:
 In mathematics, an inverse function is a function that "reverses" another function: if the function \[f\] applied to an input x gives a result of y, then applying its inverse function \[g\] to y gives the result x, i.e., \[g(y) = x\] if and only if \[f(x) = y\]. The inverse function of \[f\] is also denoted as \[{f^{ - 1}}\]. Now consider the given function \[f(x) = {3^{2x}}\]
As we know that \[f(x) = y\], on substituting it we have
\[y = {3^{2x}}\]
In RHS the number is in the form of exponential we take \[{\log _3}\]on both sides. Taking log on the both sides we have
\[{\log _3}y = {\log _3}\left( {{3^{2x}}} \right)\]
As we know the property of logarithmic function \[\log {m^n} = n\log m\], applying the property to the above function we have
\[{\log _3}y = 2x.{\log _3}\left( 3 \right)\]
The value of \[{\log _3}\left( 3 \right) = 1\], substituting in the above equation we have
\[{\log _3}y = 2x \times 1\]
On simplification
\[{\log _3}y = 2x\]
Now swap the variables that is y to x and x to y we have
\[{\log _3}x = 2y\]
Divide the above equation by 2 we get
\[ \dfrac{{{{\log }_3}x}}{2} = y\]
Therefore we have \[y = \dfrac{{{{\log }_3}x}}{2}\]. We can verify by considering the example.Consider \[f(x) = {3^{2x}}\], now take x as 1. The value is \[f(1) = {3^2} = 9\].Now consider \[{f^{ - 1}}(x) = \dfrac{{{{\log }_3}x}}{2}\], now take x as 9 then the value is
\[{f^{ - 1}}(9) = \dfrac{{{{\log }_3}9}}{2} \\
\Rightarrow{f^{ - 1}}(9)= \dfrac{{{{\log }_3}{3^2}}}{2} \\
\Rightarrow{f^{ - 1}}(9)= \dfrac{{2{{\log }_3}3}}{2} \\
\therefore{f^{ - 1}}(9)= 1\]
Hence verified.

Therefore the inverse of \[f(x) = {3^{2x}}\] is \[{f^{ - 1}}(x) = \dfrac{{{{\log }_3}x}}{2}\].

Note: In this question we must know about the logarithmic functions as we know that the logarithmic function and exponential function are inverse of each other. While using the logarithmic functions we must know the properties of logarithmic function. We must know about the simple arithmetic operations.