Question

If $f\left( x \right)={{e}^{x}}$ and $g\left( x \right)=\log \left( {{e}^{x}} \right)$ then which of the following is correct?
(a) $f\left( g\left( x \right) \right)\ne g\left( f\left( x \right) \right)$
(b) $f\left( g\left( x \right) \right)=g\left( f\left( x \right) \right)$
(c) $f\left( g\left( x \right) \right)+g\left( f\left( x \right) \right)=0$
(d) $f\left( g\left( x \right) \right)-g\left( f\left( x \right) \right)=1$

Answer 1

Simplify the function $g\left( x \right)=\log \left( {{e}^{x}} \right)$ by using the formulas $\log \left( {{a}^{m}} \right)=m\log a$ and $\log e=1$. Assume the base of the log equal to e, i.e. it is a natural log. Now, find the composite functions $f\left( g\left( x \right) \right)$ and $g\left( f\left( x \right) \right)$ one by one. To find $f\left( g\left( x \right) \right)$ substitute the value of $g\left( x \right)$ in place of $x$ in the function $f\left( x \right)$ and similarly find $g\left( f\left( x \right) \right)$ by substituting $f\left( x \right)$ in place of $x$ in the function $g\left( x \right)$. Check for the correct option.

Complete step by step solution:
Here we have been provided with the functions $f\left( x \right)={{e}^{x}}$ and $g\left( x \right)=\log \left( {{e}^{x}} \right)$, we are asked to choose the correct option regarding the composite functions $f\left( g\left( x \right) \right)$ and $g\left( f\left( x \right) \right)$. First let us simplify $g\left( x \right)=\log \left( {{e}^{x}} \right)$ further.
Now, we haven’t been provided with any base value of the log in the function $g\left( x \right)$, so in mathematics we assume the base value in such a case equal to e. In other words we can say that the given log is assumed to be a natural log. Using the formulas $\log \left( {{a}^{m}} \right)=m\log a$ and $\log e=1$ we get,
$\begin{aligned} & \Rightarrow g\left( x \right)=x\log \left( e \right) \\\ & \Rightarrow g\left( x \right)=x \\\ \end{aligned}$
The composite function $f\left( g\left( x \right) \right)$ can be obtained by substituting $g\left( x \right)$ in place of $x$ in the function $f\left( x \right)={{e}^{x}}$, so we get,
$\begin{aligned} & \Rightarrow f\left( g\left( x \right) \right)=f\left( x \right) \\\ & \Rightarrow f\left( g\left( x \right) \right)={{e}^{x}} \\\ \end{aligned}$
Similarly, the composite function $g\left( f\left( x \right) \right)$ can be obtained by substituting $f\left( x \right)$ in place of $x$ in the function $g\left( x \right)=x$, so we get,
$\begin{aligned} & \Rightarrow g\left( f\left( x \right) \right)=g\left( {{e}^{x}} \right) \\\ & \Rightarrow g\left( f\left( x \right) \right)={{e}^{x}} \\\ \end{aligned}$
Clearly we can see that the two composite functions $f\left( g\left( x \right) \right)$ and $g\left( f\left( x \right) \right)$ are equal.
$\therefore f\left( g\left( x \right) \right)=g\left( f\left( x \right) \right)$
Hence, option (b) is the correct answer.

Note: Even if we assume the base of the log equal to 10, i.e. the log is a assumed to be a common log, we are going to get the same result but then the composite functions will become $f\left( g\left( x \right) \right)=g\left( f\left( x \right) \right)={{e}^{x}}{{\log }_{10}}e$. However, always remember that if no base value is present then assume the given log as a natural log. Remember the properties of logarithms for simplifying the functions.