Prove that if a^x = b^y = (ab)^xy, then x+y = 1.

Please show me how to do this.

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- January 5th 2008, 02:36 AMgetonAnother problem in exponential and logarithms
Prove that if a^x = b^y = (ab)^xy, then x+y = 1.

Please show me how to do this. - January 5th 2008, 02:57 AMearboth
- January 5th 2008, 03:00 AMSean12345
I believe there are a few ways to do this.

- January 5th 2008, 07:48 AMSoroban
Hello, geton!

Sean is right . . . There must be dozens of approaches.

Quote:

Prove that if , then: .

We have: .

Take logs: .

* Divide by .[1]

We have: .

Take logs: .

* Divide by .[2]

Add [1] and [2]: .

. . Therefore: .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

*

If , we have: .

There are two possibilities: .

In case (1),

In case (2), is strictly determined which contradicts the original equation

. . which is true for__all__values of

Hence: .

Similarly, we can show that: .