log problem

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• Aug 28th 2008, 06:35 PM
jtc4zH
log problem
How does the following log transformation work?

(3/2)^(log2 n) = (2)^((log2 n)(log2 3/2))

I think I'm just missing something little here, but I can't think of what...

Sorry about the math, I couldn't get the math tags to work.
• Aug 28th 2008, 06:56 PM
Soroban
Hello, jtc4zH!

There may be other ways . . . possibly simpler.

Quote:

How does the following log transformation work?

$\displaystyle \left(\frac{3}{2}\right)^{\log_2(n)} \;=\;2^{\log_2(n)\cdot\log_2(\frac{3}{2})}$

$\displaystyle \text{Let: }\;x \;=\;\left(\frac{3}{2}\right)^{\log_2(n)}$

$\displaystyle \text{Take logs (base 2): }\;\log_2(x) \;=\;\log_2\!\left(\frac{3}{2}\right)^{\log_2(n)} \;=\;\log_2(n)\cdot \log_2\!\left(\frac{3}{2}\right)$

$\displaystyle \text{Exponentiate: }\;\underbrace{2^{\log_2(x)}}_{\text{This is }x} \;=\;2^{\log_2(n)\cdot\log_2(\frac{3}{2})}$

$\displaystyle \text{Therefore: }\;\left(\frac{3}{2}\right)^{\log_2(n)} \;=\;2^{\log_2(n)\cdot\log_2(\frac{3}{2})}$