Question

How do you find the inverse of $y = \log \left( {x + 1} \right)$?

Answer 1

We will equate the above function with a variable as the inverse of that variable gives us the same function.
Then we will find the value of x from the equated equation. Later we will get the inverse function as $x$ will be equal to the inverse function in $y$.
Formula Used: - If a function $f\left( x \right) = y$ , then it implies the inverse function also:
$x = {f^{ - 1}}\left( y \right)$

Complete step by step answer:
Let's say the above function is defined as $f\left( x \right) = y$.
Then the inverse of the function would be ${f^{ - 1}}\left( y \right) = x$.
But it is given that, $y = \log \left( {x + 1} \right)$.
As we know log without base notation is always considered as base 10.
Now, take 10 as base on both sides of the equation,
$\Rightarrow {10^y} = {10^{\log \left( {x + 1} \right)}}$
Now, simplify the terms,
$\Rightarrow {10^y} = x + 1$
On subtracting 1 from both sides, we get
$\Rightarrow {10^y} - 1 = x + 1 - 1$
Simplify the terms,
$\Rightarrow {10^y} - 1 = x$
Now, if we replace the value of $y$ by $x$ and $x$ by $y$ then we can say that,
$\therefore y = {10^x} - 1$

Hence, the inverse of $y = \log \left( {x + 1} \right)$ is $y = {10^x} - 1$.

Note: The inverse of a function is a function that is reverse or reciprocal of that function.
If the function $f$ applied to an input $x$ gives a result of $y$, then applying its inverse function $g$ to $y$ will give us the result of $x$.
Always remember that the inverse of a function is denoted by ${f^{ - 1}}$.
Some properties of a function are given below:
There is an always symmetry relationship exists between function and its inverse, that is why it states:
${\left( {{f^{ - 1}}} \right)^{ - 1}} = f$
If an inverse function exists for a given function then it must be unique by its property.