Question

What is the inverse of $y={{6}^{x}}$?

Answer 1

To find the inverse of the given exponential function, first of all find the value of x as a function of y. To do this, take log to the base 6 both the sides and use the formulas $\log {{a}^{m}}=m\log a$ and ${{\log }_{a}}a=1$, to simplify the R.H.S. Now, assume $y=f\left( x \right)$ and write $x={{f}^{-1}}\left( y \right)$. Substitute the obtained value of x in terms of y in this assumed relation and finally replace y with x to get the inverse function ${{f}^{-1}}\left( x \right)$.

Complete step-by-step solution:
Here we have been provided with the exponential function $y={{6}^{x}}$ and we are asked to find the inverse of this function.
Now, we can see that the given function is an exponential function so its inverse must be a logarithmic function.
$\because y={{6}^{x}}$
Taking log to the base 6 both the sides we get,
$\Rightarrow {{\log }_{6}}y={{\log }_{6}}\left( {{6}^{x}} \right)$
Using the property of log given as $\log {{a}^{m}}=m\log a$ we get,
$\Rightarrow {{\log }_{6}}y=x{{\log }_{6}}\left( 6 \right)$
In the R.H.S we have the base and the argument of the log equal to each other so we can use the formula ${{\log }_{a}}a=1$ we get,
$\begin{aligned} & \Rightarrow {{\log }_{6}}y=x\times 1 \\\ & \Rightarrow x={{\log }_{6}}y \\\ \end{aligned}$
Now, the given expression was $y={{6}^{x}}$ that is y as a function of x so we can write it as $y=f\left( x \right)$. Taking ${{f}^{-1}}$ both the sides we get,
$\Rightarrow x={{f}^{-1}}\left( y \right)$
Substituting the logarithmic expression of x in terms of y obtained above we get,
$\Rightarrow {{f}^{-1}}\left( y \right)={{\log }_{6}}y$
Substituting x in place of y in the above relation we get,
$\Rightarrow {{f}^{-1}}\left( x \right)={{\log }_{6}}x$
Hence, the above relation is our answer.

Note: Always remember that the logarithmic function and exponential function are inverse functions of each other by definition. Often we convert an exponential function into log function for the calculations in mathematics with the help of log tables. Do not forget to replace y with x in the end because we always have to find ${{f}^{-1}}\left( x \right)$ and not ${{f}^{-1}}\left( y \right)$. If you want you can use the base change rule of log given as ${{\log }_{a}}b=\dfrac{{{\log }_{c}}b}{{{\log }_{a}}b}$ to convert the expression ${{\log }_{6}}x$ into any base of the log.