Question

Let f be a function defined for x>0 such that f'(x)=1/x for all x>o and f(1)=0 show that f(ab)=f(a)+f(b), f(a/b)=f(a)-f(b) and f(a^r)=rf(a) for all a,b>0, r (do not use f(x)=lnx)

Answer 1

Could not integrate f'(x) then prove?

Answer 2

could we use lnx+c

Answer 3

I would use that too if it is possible, @anonymoustwo44

Answer 4

if f'(x)=1/x then f(x) could only be \[\ln \left| x \right|\]

Answer 5

f'(x) =1/x
f(x) = ln| x |+C
f(1)=0
ln1+C=0
C=0
f(x)=ln| x|

f(ab) = ln|ab|=ln |a| +ln|b| = f(a)+f(b)
f(a/b)=ln |a/b|=ln|a|-ln|b| =f(a)-f(b)
f(a^r)=ln|a^r|=rln|a|=rf(a)

Answer 6

Will you do it like this, @anonymoustwo44

Answer 7

yes exactly :))