Another Proof question involving logs?

Question

These are really starting to frustrate me, still its the last question in my self teach book, but they never show u the friggin workings!

it says: assuming that a = e^lna (which I know is a true fact anyway) when a >0 Proove that

ln(a^n) = nlna

Now I know that alogb = logb^a. But is the above bit to throw me or what. I could proove it if I just said let b = lna^n. But the above has thrown me...again.

Any help much appreciated

I

Answer 1

You have to use the property that (a^b)^c = a^(bc)...

ln(a^n) = n ln a
ln((e^(ln a))^n) = n ln a
ln(e^(n ln a)) = n ln a
n ln a = n ln a

Answer 2

In other words use the properties:

Given a=e^ln(a)
Raise both sides to the nth power: a^n=[e^ln(a)]^n
Properties of exponents tells you to multiply exponents so:
a^n=e^(n*ln(a))
The definition of logarithms then says ln(a^n)=n*ln(a)

QED