# Thread: Prove the 4th Law of Exponents

1. ## Prove the 4th Law of Exponents

Here is what I did, but I'm not sure that this is valid:

(ab)^x = e^{x ln(ab)} = e^{x ln(a) + x ln(b)} = e^{x ln(a)}e^{x ln(b)} = a^xb^x

What do you think?

Thanks!

2. I assume you would not be able to use logarithm laws until you have proven all of the index laws...

I would say...

3. That's a good idea. They show us how they prove the first two laws of exponents, and they use logarithms to do it. My attempt is a variation on what the book did for the first two proofs.

I'll try your way, too.

Thanks!

4. Prove It's argument only works if x is a positive integer. A more formal proof (for x a positive integer) would require mathematical induction.

For x an arbitrary real number, the OP's method can be used.

5. Originally Posted by DrSteve
Prove It's argument only works if x is a positive integer. A more formal proof (for x a positive integer) would require mathematical induction.

For x an arbitrary real number, the OP's method can be used.
Assuming of course that logarithm rules can be used...

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# fourth law of logarithms

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