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Simplify: $$\frac{\log a + \log b - \log c}{\log d^2}$$

Using the basic properties of logs, the numerator should simplify to $\log (ab/c)$, if I'm not mistaken. The denominator $\log d^2 = 2 \log d$ but I don't know where to go from there. Can it be further simplified?

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  • $\begingroup$ you can go any farther; answer is $\frac{\log(ab/c)}{2\log d}.$ $\endgroup$ – abel May 14 '15 at 2:34
  • $\begingroup$ Eh. You could do change of base, but arguably that makes it worse. $\endgroup$ – jgon May 14 '15 at 2:39
  • $\begingroup$ So the best answer here is probably just the simplification of the numerator as described above? $\endgroup$ – Lulu Uy May 14 '15 at 2:43
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You may further simplify this to: $log_d \sqrt{\frac{ab}{c}}$ As Jgon suggested.

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  • $\begingroup$ How exactly does that work, I don't understand ? $\endgroup$ – Lulu Uy May 14 '15 at 2:59
  • $\begingroup$ $log{a}/log{b}=log_b{a}$ I leave it to you to prove. Good exercise for practice. $\endgroup$ – Moti May 14 '15 at 3:00
  • $\begingroup$ I understand this now, but how do you turn the d^2 into the square root of ab/c ? $\endgroup$ – Lulu Uy May 14 '15 at 3:11
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    $\begingroup$ Don't square the d, use the 2 that you get from $logd^2 =2logd$ to divide the numerator.. $\endgroup$ – Moti May 14 '15 at 3:23

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