Question

You are supposed to find the inverse of $y = {e^x}$

Answer 1

We are given an exponential function on the left hand side of the equation and we have to find its inverse i.e. logarithmic function. For this we use the conversion of exponential-logarithmic functions because both are inverse functions of each other.

Complete solution step by step:
Firstly we write the given equation and name it as equation (1)
$y = {e^x}\,{\text{ - - - - - - - - (1)}}$
The right hand side of equation (1) is a natural exponential function consisting of $e$ which is the exponential constant having an approximate value of 2.718 and we also know that
Exponential function is the Inverse function of a logarithmic function. This means one can be undone or removed by operating the other function on it and vice versa. So the inverse of natural exponential function will be natural logarithmic function. For that we take natural logarithm ‘ln’ on both sides of equation (1) because it also has $e$ as its base, and obtain this –
${\log _e}(y) = {\log _e}({e^x}) \\\ \Rightarrow \ln (y) = \ln ({e^x})\,\,\\{ \because {\log _e}({e^x}) = \ln ({e^x})\\} \\\$
By the definition of natural logarithm, we know

$$[\ln c = b] \Leftrightarrow [{e^b} = c] \\\ \Rightarrow \ln ({e^x}) = x \\\$$

Putting the value in the above equation we have
$\ln (y) = \ln ({e^x}) \\\ \Rightarrow \ln (y) = x \\\$
This is the logarithmic form of the given expression which is the inverse of the given problem.
Additional information: The property which has been used here is-
${a^b} = c \\\ {\log _a}c = {\log _a}({a^b}) = b \\\$
This helps us to understand the reason why they are inverse functions of each other.

Note: The natural exponential constant used here is known as Euler’s number. Its properties have led it to be a natural choice of logarithmic base. The differentiation of ${e^x}$ comes out to be the same as ${e^x}$ making it unique among the other functions.