proof of derivative of a^x

how do you prove that the derivative of a^x = (a^x)(ln a)?

i tried using logarithmic differentiation:

y = a^x

ln y = ln (a^x)

ln y = x (ln a)

d/dx (ln y) = d/dx [x (ln a)]

(1/y) (dy/dx) = (1)(ln a) + (x)(1/a)

dy/dx = y [(ln a) + (x/a)]

so if i substitute

dy/dx = a^x [(ln a) + (x/a)]

did i do something wrong? how come it's not coming out correctly? i used logarithmic differentiation to prove that the derivative of e^x = e^x and it worked. how come it doesn't work for proving this?