# How do you evaluate log_144 12?

Sep 8, 2016

$\frac{1}{2}$

#### Explanation:

${\log}_{144} 12 = {\log}_{e} \frac{12}{{\log}_{e} 144} = {\log}_{e} \frac{12}{{\log}_{e} {12}^{2}} = \frac{1}{2}$

Sep 8, 2016

${\log}_{144} 12 = \frac{1}{2}$

#### Explanation:

${\log}_{144} 12$
Written in this form, the question that is being asked is:

"What index of 144 will give 12?"
Or" How can I get from 144 to 12 by using an index/power?"

You should recognise that $\sqrt{144} = 12$

Written in index form, this is ${144}^{\frac{1}{2}} = 12$

${\log}_{144} 12 = \frac{1}{2}$

If the answer was not an exact value this can be calculated using the change of base as explained by another contributor.

Knowing the smaller powers by heart allows you to write answers immediately.