Question

How do you find the inverse of $y = {3^x}$?

Answer 1

Hint: We can observe that these types of functions can be solved only by using logarithmic equations. We have a variable, which is x in the question, as a power to a constant which is 3.

Complete step by step solution:
According to the question, we have to find the inverse of the equation $y = {3^x}$,
Whenever we have to find the inverse of any function we have to replace the variable x with y and variable y with x. Then we have to form a function of y equal to a function of x. This new equation will be the inverse that we need.
So, our new function will be
$ \Rightarrow x = {3^y}$
So at first, we have to take log on both sides, but we have to take the log with a base of 3, so we will get
$ \Rightarrow {\log _3}(x) = {\log _3}({3^y})$
Now, as we know that when we have any term inside a log function with any power, the power will come down and get multiplied with the log of that term. So we will get our equation as,
$ \Rightarrow {\log _3}(x) = y{\log _3}(3)$
And we know that log of any number is equal to one of the bases is the same as that number. So the value of ${\log _3}(3)$ is equal to one, hence our equation will become.

$ \Rightarrow y = {\log _3}(x)$

Note: Basically whenever we rake log we take it in base ten. But in this question we have a constant that is equal to three, so we took base three to make it unity and simplify our calculation. But we can take logS with any possible value as a base, but our calculation will be tough.