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

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- Jul 11th 2010, 05:36 AMMissProve a natural logarithmic property using ... ?
__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 - Jul 11th 2010, 07:16 AMFailure
- Jul 11th 2010, 07:25 AMMiss
Thanks

but you have a small mistake

the lower limit for integral should be 1 not 0

thanks again - Jul 11th 2010, 07:27 AMFailure
- Jul 11th 2010, 07:34 AMHallsofIvy
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 . - Jul 11th 2010, 07:58 AMPlato
Another way.

Here is the way Leonard Gillman does this problem.

Because we use .

Using u-substitution .

Now consider that - Jul 11th 2010, 08:11 AMMiss
HallsOfIvy :

Sir, you should preview your reply before post it

there are 45545 replies for you with latex errors and non-complete latex codes :p

Thanks

Plato:

Thanks