Question about logarithms

Hi Guys,

I having some trouble with logarithms again. I am able to solve them okay most of the time. But my final answer answer doesn't match the given one.

Take the following problem for instance. I solved upto

$\displaystyle

log_{\sqrt{12}} 4 + log_{12} \sqrt{4}

$

$\displaystyle

= 2log_{\sqrt{12}} 2 + log_{12} 2

$

$\displaystyle

= \dfrac{2}{log_2 \sqrt{12}} + \dfrac{1}{log_2 12}

$

$\displaystyle

= \dfrac{2}{\frac{1}{2} {log_2 12}} + \dfrac{1}{log_2 12}

$

$\displaystyle

= \dfrac{4}{log_2 12} + \dfrac{1}{log_2 12}

$

$\displaystyle

= \dfrac{5}{log_2 12}

$

The required answer is,

$\displaystyle

\frac{5}{2}log_{12} 4

$

In this case I see that I could get to the required answer by going flipping the base, 4^1/2 and so on.

In this example I am close to the final answer, but many times I am not this close. Is there a general rule to follow when simplifying? How do I figure out what base the answer should be in? Where should the simplification end?