Question

Find the inverse of the function$y = \log \left( {x - 2} \right).$

Answer 1

Hint:We know ${\log _b}\left( P \right) = Q\;\; \Rightarrow P = {b^Q}$ since the exponential and logarithmic functions are inverse in nature. It converts the logarithmic equation into its equivalent exponential form. So by using the above identity we would be able to find the inverse of the function$y = \log \left( {x - 2} \right)$.

Complete step by step answer:
Given, $y = \log \left( {x - 2} \right)......................................\left( i \right)$
Now to find the inverse of the given logarithmic functions the following steps are to be followed:
1. Interchange the variables $x\;{\text{and}}\;y$with each other.
2. Isolate the expression containing logarithmic terms on one side of the equation and all other terms towards the other side.
3. Using ${\log _b}\left( P \right) = Q\;\; \Rightarrow P = {b^Q}$ converts the logarithmic equation into its equivalent exponential form.
4. In order to get the inverse, solve the exponential equation for y and then replace\[y\;{\text{by}}\;{f^{ - 1}}\left( x \right)\].

Now let’s find the inverse of $y = \log \left( {x - 2} \right).$First interchanging the variables $x\;{\text{and}}\;y$ with each other, we get:
$x = \log \left( {y - 2} \right)......................\left( {ii} \right)$
Here we can see that the logarithmic terms and other terms are separated already so we need not to separate it.Now we need to convert the logarithmic equation to exponential equation using the equation:
${\log _b}\left( P \right) = Q\;\; \Rightarrow P = {b^Q}$
So on applying it to (ii) we get:
$
x = \log \left( {y - 2} \right) \\
\Rightarrow y - 2 = {10^x}.......................\left( {iii} \right) \\ $
Now solving for $y$, we get:
$
y - 2 = {10^x} \\
\Rightarrow y = {10^x} + 2.................\left( {iv} \right) \\ $
Replacing\[y\;{\text{by}}\;{f^{ - 1}}\left( x \right)\], we get:
\[{f^{ - 1}}\left( x \right) = {10^x} + 2\]

Therefore the inverse of $y = \log \left( {x - 2} \right)$ is:\[{f^{ - 1}}\left( x \right) = {10^x} + 2\].

Note:Some of the logarithmic properties useful for solving questions of derivatives are listed below:
$
1.\;{\log _b}\left( {PQ} \right) = {\log _b}\left( P \right) + {\log _b}\left( Q \right) \\
2.\;{\log _b}\left( {\dfrac{P}{Q}} \right) = {\log _b}\left( P \right) - {\log _b}\left( Q \right) \\
3.\;{\log _b}\left( {{P^q}} \right) = q \times {\log _b}\left( P \right) \\ $
These are called Product Rule, Quotient Rule and Power Rule respectively.Common logarithmic functions are log functions with base 10, and natural logarithmic functions are log functions with base ‘e’. Natural logarithmic functions can be represented by $\ln x\;{\text{or}}\;{\log _e}x$.