Question

Which of the following functions are the mirror images of each other with respect to the line $y=x$

• A. $e^x$ and ${\log }_{e}x$
• B. $e^x$ and ${\log }_{x}e$
• C. ${\pi }^x$ and ${\log }_{\pi }x$
• D. ${\pi }^x$ and ${\log }_x\pi$

Answer 1

Answer: (A), (C)

The correct options are
A $e^x$ and ${\log }_{e}x$
C ${\pi }^x$ and ${\log }_{\pi }x$

We know that, graph of exponential function and logarithmic function are the mirror images of each other with respect to the line $y=x$ if their bases are same. i.e., $a^x$ and ${\log }_{a}x$


So, $e^x$ and ${\log }_{e}x$ are the mirror images of each other.
${\pi }^x$ and ${\log }_{\pi }x$ are the mirror images of each other.

${\log }_{x}e=\frac{1}{{\log }_{e}x}$
So, $e^x$ and $\frac{1}{{\log }_{e}x}$ are not the mirror images of each other.
Similarly, ${\pi }^x$ and ${\log }_x\pi$ are not the mirror images of each other.