. x^2 = 2^x 2 (ln x) = x (ln 2) (ln x)/x = (ln 2)/2 The following is the plot of the last expression, and you get two roots, 2 and 4. It is interesting to see what happens if it is not x-squared but higher orders. It seems like there are only two real roots for everything more than (raised to the power) 2 except e itself, in which case we have only one real root. But there has to be more roots for higher order, right? Like cubic expressions should have 3 roots. I guess the ones we don't see are imaginary roots? As q tends to infinity, (ln q)/q tends to 0. Also, this line (ln q)/q first increases then decreases. Of course, for q<1, we have only one root - see the plot.