2^x = 2x

ln(2^x) = ln(2x)

xln(2) = ln(2) + ln (x)

x = [ln (2) + ln (x)] / ln (2)

x = 1 + ln(x)/ln(2)

From here, by simple inspection, we can see that x can be either 1 or 2.

when x = 1, we have 1 = 1 + ln(1)/ln(2) = 1 + 0

when x = 2, we have 2 = 1 + ln(2)/ln(2) = 1 + 1