Question

How do you find the inverse of $y = \ln (x + 1)$?

Answer 1

Hint: In order to determine the inverse of the above question, we have to write x in term of y ,but for this we have to first remove the $\ln $ .So this we will be convert the equation into exponential form, and to do so use the definition of logarithm that the logarithm of the form ${\log _b}x = y$ is when converted into exponential form is equivalent to ${b^y} = x$,so compare with the given logarithm value with this form and replace the variable $x$ with $y$ and vice versa ,we will get your required answer.

Complete step by step answer:
To find the inverse of given the logarithmic equation$y = \ln (x + 1)$, we must know the properties of logarithms and with the help of them we are going to rewrite our question.
To find inverse of any equation we have to follow certain steps:
1.write the equation $x$ in terms of $y$.
2.replace the variable $x$ with $y$ and vice versa.
3. You have successfully obtained the inverse.

To write the equation $y = \ln (x + 1)$ of $x$ in terms of $y$. We have to convert it into exponential form as we know the logarithm is actually the inverse of exponential.Recall that $\ln $ is nothing but logarithm having base $e$.We can rewrite our expression as
$y = {\log _e}(x + 1)$
Let’s convert this into its exponential form to remove ${\log _e}$. Any logarithmic form ${\log _b}X = y$ when converted into equivalent exponential form results in ${b^y} = X$.So in Our question we are given ${\log _{10}}x = 4$ and if compare this with ${\log _b}x = y$ we get,
$
b = e \\
\Rightarrow y = y \\
\Rightarrow X = x + 1 \\ $
\[
\Rightarrow y = {\log _e}(x + 1) \\
\Rightarrow {e^y} = x + 1 \\
\Rightarrow x = {e^y} - 1 \\ \]
Now replacing the variable $x$with $y$ and vice versa, we get
\[ \therefore y = {e^x} - 1\]

Therefore, the inverse of the equation $y = \ln (x + 1)$ is equal to \[y = {e^x} - 1\].

Note:Don’t forget to cross check your result. $\ln $ is known as natural logarithm. Logarithm of constant 1 is equal to zero.Value of constant” e” is equal to 2.71828. A logarithm is basically the reverse of a power or we can say when we calculate a logarithm of any number , we actually undo an exponentiation.