Differentiation of logarithms and exponential......HELP!?

Question

x= base^exponent
log base x ( e.g log 4 16= 2)
lg= log base of 10
ln= log base of e

This is given in the book:
d(ln x) / dx = 1/x
d(lg x) / dx = lg e / x
d(e^ax) / dx = ae^ax

so I need to know what is d(5^ax) / dx and d(log 2 x) / dx and if they are any different from the forms I show above. Thanks.

Answer 1

log (base 2) x = ln 2 / ln x

a^x = e^(x ln a)

d(5^ax)/dx = d(e^(ax * ln 5))/dx = a*ln5*e^(ax * ln 5) =
= 5^ax * (a*ln 5)

d (log base 2 x)/dx = d((ln x) / (ln 2))/dx = 1/ln2 d(ln x)/dx =
= 1/(x ln 2)

Answer 2

I'll show you the different logarithms and exponentials:

1) f(x) = e^x
2) f(x) = a^x (for some constant "a").
3) f(x) = ln(x) (The natural log of x; base e)
4) f(x) = log[base a](x).

The derivative of e^x is simply itself, e^x.
The derivative of a^x would be (a^x)ln(a).
The derivative of ln(x) would be (1/x).
The derivative of log[base a](x) would be 1/[x ln(a)].