Curve sketching practice question 24

The solutions are where the curve $y = e^{x}$ meets the line $y = 3x$. Can you recast that as the zeros of a single function, and let its derivative tell you how many there can be?

Count the zeros of $g(x) = e^{x} - 3x$ instead. Where is $g$ decreasing, where is it increasing, and what is its smallest value?

$g'(x) = e^{x} - 3$ is zero at $x = \log 3$. Compare $g(\log 3) = 3 - 3\log 3$ with 0, using $\log 3 > \log e = 1$, and find the limits of $g$ as $x \to \pm\infty$.