The Question:
Use the definition of the natural logarithm function to prove that :
ln (ab) = ln(a) + ln(b)
for any positive number a and b.
my problem is that I do not know how to prove it using the definition
BTW, the definition is
A slight variation, a bit more direct, I think, since it uses the integral only and not the derivative:
If a is positive, then is also positive and, by that definition, .
Let u= at so that t= u/a, dt= (1/a)du. When t= 1, u= a, when t= 1/a, u= 1. Now we have . The "a"s cancel giving
.
If a and b are positive then ab is also positive and, by that definition,
.
Now, let u= t/a so that t= au, dt= adu, (1/t)dt= (1/au)(adu)= (1/u)du. When t= 1, u= 1/a and when t= ab, u= b. The integral becomes .