Question

How do you simplify the given function ${{e}^{\ln \left( 5x+2 \right)}}$?

Answer 1

We start solving the problem by making use of the fact that the function $\ln \left( g\left( x \right) \right)$ defines only if $g\left( x \right)>0$. We use this fact and make the necessary calculations to get the domain of the given function. We then make use of the fact that ${{e}^{\ln \left( g\left( x \right) \right)}}=g\left( x \right)$, for all the values of x present in the domain of the given function to get the required simplified form of the given function.

Complete step by step answer:
According to the problem, we are asked to simplify the given function ${{e}^{\ln \left( 5x+2 \right)}}$.
Let us assume $f\left( x \right)={{e}^{\ln \left( 5x+2 \right)}}$ ---(1).
We know that the function $\ln \left( g\left( x \right) \right)$ defines only if $g\left( x \right)>0$. Let us use this definition for the function in equation (1).
$\Rightarrow 5x+2>0$.
$\Rightarrow 5x>-2$.
$\Rightarrow x>\dfrac{-2}{5}$.
$\Rightarrow x\in \left( \dfrac{-2}{5},\infty \right)$.
So, the domain of the given function $f\left( x \right)={{e}^{\ln \left( 5x+2 \right)}}$ is $\left( \dfrac{-2}{5},\infty \right)$.
Now, let us simplify the given function $f\left( x \right)={{e}^{\ln \left( 5x+2 \right)}}$ ---(2).
We know that ${{e}^{\ln \left( g\left( x \right) \right)}}=g\left( x \right)$, for all the values of x present in the domain of the given function. Let us use this result in equation (2).
$\Rightarrow f\left( x \right)=5x+2$, for $x\in \left( \dfrac{-2}{5},\infty \right)$.
So, we have found the simplified form of the given function ${{e}^{\ln \left( 5x+2 \right)}}$ as $5x+2$, for $x\in \left( \dfrac{-2}{5},\infty \right)$.
$\therefore$ The simplified form of the given function ${{e}^{\ln \left( 5x+2 \right)}}$ is $5x+2$, for $x\in \left( \dfrac{-2}{5},\infty \right)$.

Note:
We should keep in mind that the given function cannot be simplified for all real values of x i.e., $x\in \mathbb{R}$, which is the common mistake done by students. Whenever we get this type of problem, we first find the domain of the given function. We should report the domain of the given function as $\left[ \dfrac{-2}{5},\infty \right)$ which is another mistake done by students. Similarly, we can expect problems to find the range of the given function: ${{e}^{\ln \left( 5x+2 \right)}}$.