Ssimplifying Logs

**Mr_Green** · November 7th 2006, 03:55 PM

log base 5 of (8) * log base 64 of (25)

Could someone show me the steps so I can apply it to my other hw problems.

Thanks,
Mr_GREEN

**Soroban** · November 7th 2006, 04:13 PM

Hello, Mr_Green!

Are you allowed to use the Base-Change Formula?

Simplify: . $\log_5(8)\cdot\log_{64}(25)$

Using the Base-Change Formula, we have: . $\frac{\log(8)}{\log(5)}\cdot\frac{\log(25)}{\log(6 4)}$

. . $= \:\frac{\log(8)}{\log(5)}\cdot\frac{\log(5^2)}{\lo g(8^2)} \:=\:\frac{\log(8)}{\log(5)}\cdot\frac{2\!\cdot\!\ log(5)}{2\!\cdot\!\log(8)} \:=\:1$

**Quick** · November 7th 2006, 06:02 PM

Originally Posted by Soroban

Hello, Mr_Green!

Are you allowed to use the Base-Change Formula?

Using the Base-Change Formula, we have: . $\frac{\log(8)}{\log(5)}\cdot\frac{\log(25)}{\log(6 4)}$

. . $= \:\frac{\log(8)}{\log(5)}\cdot\frac{\log(5^2)}{\lo g(8^2)} \:=\:\frac{\log(8)}{\log(5)}\cdot\frac{2\!\cdot\!\ log(5)}{2\!\cdot\!\log(8)} \:=\:1$

Wow, yet another convenient thing with logs. Does it convert the numbers to log_10?

**Soroban** · November 7th 2006, 06:23 PM

Originally Posted by Quick

Does it convert the numbers to log_10?

It converts to any base: . $\log_a(N) \;=\;\frac{\log_b(N)}{\log_b(a)}$

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