Question

How do you find the inverse of $y={{\log }_{8}}\left( x+2 \right)$?

Answer 1

To solve the given equation we will take the help of the formula ${{\log }_{a}}x=y\Leftrightarrow {{a}^{y}}=x$. Sine, the inverse of y here will be x therefore, after solving it we will solve for x. For solving we will replace a as 8 and x as x + 2. Finally we are going to substitute these into the formula and finally we will shift x and y with each other’s positions to get the required answer.

Complete step by step answer:
We will consider $y={{\log }_{8}}\left( x+2 \right)$. As we know that ${{\log }_{a}}x=y\Leftrightarrow {{a}^{y}}=x$ therefore, by taking a as 8 and x as x + 2 we get ${{8}^{y}}=x+2$.
As we know that the inverse of y will be x in equation $y={{\log }_{8}}\left( x+2 \right)$ so, we will solve ${{8}^{y}}=x+2$ further and get ${{8}^{y}}-2=x$ by taking 2 to the left side of equal sign.
Now, we will interchange the variables x and y with each other and get the inverse of the function $y={{\log }_{8}}\left( x+2 \right)$ as ${{8}^{x}}-2=y$.

Hence, the required inverse is ${{8}^{x}}-2=y$.

Note: It is important to get the inverse by interchanging x and y with each other. But we need to use the process of interchanging variables x and y with each other at the right time. If we interchange them at first place, then the solution would have been as follows.
We will consider $y={{\log }_{8}}\left( x+2 \right)$ and interchange the positions of x and y to get $x={{\log }_{8}}\left( y+2 \right)$. As we know that ${{\log }_{a}}x=y\Leftrightarrow {{a}^{y}}=x$ therefore, by taking a as 8 and x as y + 2 we get ${{8}^{x}}=y+2+2$. This gives us the wrong answer. This question needs a lot of concentration because if we use the formula without focusing on it then we might lead towards the wrong answer and our hard work will become zero. Any question of $y={{\log }_{8}}\left( x+2 \right)$ type will need formula ${{\log }_{a}}x=y\Leftrightarrow {{a}^{y}}=x$ to be solved further.