Q.Compare the graphs of $y=2^x$ and $y=\log_2 x$, stating the domain, range, and the relationship between the two graphs.

These two functions are inverses of each other with the same base, so their domains and ranges swap, and their graphs are mirror images of one another.

Step 1 — $y=2^x$. Any real number can be used as an exponent, so the domain is $\mathbb{R}$. Since $2^x>0$ for every real $x$ (a positive base raised to any power stays positive) and $2^x$ takes every positive value as $x$ ranges over $\mathbb{R}$, the range is $(0,\infty)$. The graph passes through $(0,1)$ (since $2^0=1$), rises steeply for large $x$, and approaches (but never reaches) the $x$-axis as $x\to-\infty$.

Step 2 — $y=\log_2x$. By definition, $\log_2x$ asks 'to what power must $2$ be raised to give $x$?', which only makes sense for $x>0$ — domain $(0,\infty)$. As $x$ ranges over all positive numbers, the required power ranges over all real numbers — range $\mathbb{R}$. The graph passes through $(1,0)$ (since $2^0=1$), rises slowly, and has a vertical asymptote at $x=0$. …

Mixing up which function has which domain/range — remembering that an exponential's output is always positive (so that becomes the log's input restriction) helps avoid swapping them incorrectly. Some students also forget the …