Properties of exponential function to solve $3x=e^x$ Solve $ {3}{x}\mathrm{{=}}{e}^{x} $ Solving graphically with the help of a grapher tool we get $x\approx 1.8$ and approx $4.5$. Which properties of exponential functions tells us that there will be two solutions? Are there always two solutions when a (linear function )= an (exponential function)?

Let $f(x) = 3x - e^x$. Then $f(0) = -1$ and $f(1) = 3 - e > 0$, so by the IVT $f$ has a zero in $(0,1)$. But $\lim_{x\to\infty}f(x) = -\infty$, so there must be another zero in $(1,\infty)$. On the other hand, $f(x) < 0$ for all $x < 0$, so there are no zeros in $(-\infty,0)$. Thus $f$ has exactly two zeros.

There will not always be two solutions. Consider $x = e^x$ and $x + 1 = e^x$.