## Simplifying a Log Equation

Complex numbers, rational functions, logarithms, sequences and series, matrix operations, etc.
GreenLantern
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Joined: Sat Mar 07, 2009 10:47 pm
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### Simplifying a Log Equation

My instructions are simple, "Write as a sum and difference of the simplest logarithms possible."

$log_3((x+5)^2 \sqrt{\frac{(2x+3)}{(x-2)^3} })$

My attempts have lead me down the following...

$log_3((x+5)^2\sqrt{\frac{(2x+3)}{(x-2)^3} })$ Base problem
$log_3((x+5)^2(\frac{(2x+3)}{(x-2)^3})^{\frac{1}{2}\ })$ Removed root
$log_3((x+5)^2(2x+3)^{\frac{1}{2}}(x-2)^{\frac{2}{3}})$ Simplified fraction
$log_3(x+5)^2+log_3(2x+3)^{\frac{1}{2}}+log_3(x-2)^{\frac{2}{3}}$ No idea what this is but I know it works

... and I think that's all?

stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
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My instructions are simple, "Write as a sum and difference of the simplest logarithms possible."

$log_3\left((x+5)^2 \sqrt{\frac{(2x+3)}{(x-2)^3} }\right)$

My attempts have lead me down the following...
As long as there are powers inside the log, you aren't done, because of the log rule that lets you take those out front as multipliers.

It might be simpler to deal with the radical (that is, the one-half power) earlier on:

. . . . .$\log_3\left((x\, +\, 5)^2\right)\, +\, \log_3\left(\sqrt{\frac{(2x\, +\, 3)}{(x\, -\, 2)^3}}\right)$

. . . . .$2\times \log_3(x\, +\, 5)\, +\, \left(\frac{1}{2}\right)\times\log_3\left(\frac{(2x\, +\, 3)}{(x\, -\, 2)^3}\right)$

. . . . .$2\times \log_3(x\, +\, 5)\, +\, \left(\frac{1}{2}\right)\times\log_3(2x\, +\, 3)\, -\, \left(\frac{1}{2}\right)\times\log_3\left((x\, -\, 2)^3\right)$

Then handle the cube inside the last log similarly.