Question related to the mean value theorem and increasing and decreasing functions

Question

Let L be a differentiable function on (0, ∞) such that L'(x)=1/x and L(1) = 0. Prove that for any two positive numbers a and b, L(ab) = L(a) + L(b). HINT: Begin by showing that, for any x > 0, L(ax) and L(x) have the same derivative.

How can I find L(x) ?

Answer 1

Dear Paw,

For any x>0 and a>0 we have L' (ax) = (1/ax) (ax)' = (1/ax) a =1/x.

That is L'(x) =L'(ax)

which implies that L(ax) and L(x) differ by a constant ,say k.

Then L(ax) = L (x) + k Equation (1) .

Now plug in x=1 into Equation (1)

L( a1) = L(1) +k

L(a) = 0 +k

L(a)=k

Therefore L(ax) = L (x) +L(a).

This is the famous Logarithmic function