Prove the 4th Law of Exponents

• Apr 17th 2011, 09:24 PM
joatmon
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!
• Apr 17th 2011, 09:40 PM
Prove It
I assume you would not be able to use logarithm laws until you have proven all of the index laws...

I would say...

http://quicklatex.com/cache3/ql_624c...91bc415_l3.png
• Apr 17th 2011, 09:48 PM
joatmon
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.

Thanks!
• Apr 18th 2011, 06:25 AM
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.
• Apr 18th 2011, 02:54 PM
Prove It
Quote:

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...