Question

How do you write $y=\ln \left( x-2 \right)$ in exponential form?

Answer 1

Hint: In this question, we have to find the exponential form of the given logarithmic equation. Thus, we will use the exponential formula and the mathematical rule to get the solution. First, we will put the exponential function on both sides in the equation. Then, we will apply the log-exponential formula ${{e}^{\log a}}=a$ on the right hand side of the equation. In the last, we will add 2 on both sides of the equation and make the necessary calculations, to get the required solution for the problem.

Complete step by step solution:
According to the problem, we have to find the exponential form of the given logarithmic equation.
The logarithmic equation given to us is $y=\ln \left( x-2 \right)$ -------- (1)
First, we will put the exponential function on both sides in the equation (1), we get
$\Rightarrow \exp \left( y \right)=\exp \left( \ln \left( x-2 \right) \right)$
Therefore, we get
$\Rightarrow {{\operatorname{e}}^{y}}={{\operatorname{e}}^{\ln \left( x-2 \right)}}$
Now, we will apply the log-exponential formula ${{e}^{\log a}}=a$ on the right hand side of the above equation.
$\Rightarrow {{\operatorname{e}}^{y}}=x-2$
Now, we will add 2 on both sides in the above equation, we get
$\Rightarrow {{\operatorname{e}}^{y}}+2=x-2+2$
As we know, the same terms with opposite signs cancel out each other, thus we get
$\Rightarrow {{\operatorname{e}}^{y}}+2=x$ which is answer
Therefore, for the equation $y=\ln \left( x-2 \right)$ , its exponential form of equation is equal to ${{\operatorname{e}}^{y}}+2=x$.

Note: While solving this problem, do mention all the steps properly to avoid confusion and mathematical error. One of the alternative methods to solve this problem is take the antilog on both sides of the equation and make the necessary calculations, to get the accurate answer.