# How do I solve log_0.25[ (log_2 3)(log_3 4)] ?

Oct 14, 2017

The answer is $- \frac{1}{2}$.

#### Explanation:

Start by rewrite $0.25$ as $\frac{1}{4}$.

$= {\log}_{\frac{1}{4}} \left[\left({\log}_{2} 3\right) \left({\log}_{3} 4\right)\right]$

We can now use the change of base formula, which states that ${\log}_{a} n = \log \frac{n}{\log} a$.

$= {\log}_{\frac{1}{4}} \left[\frac{\log 3}{\log 2} \cdot \log \frac{4}{\log} 3\right]$

$= {\log}_{\frac{1}{4}} \left(\log \frac{4}{\log} 2\right)$

$= {\log}_{\frac{1}{4}} \left(\log \frac{{2}^{2}}{\log} 2\right)$

$= {\log}_{\frac{1}{4}} \left(\frac{2 \log 2}{\log} 2\right)$

$= {\log}_{\frac{1}{4}} 2$

Once again we use the change of base formula.

=log2/(log(1/4)

$= \log \frac{2}{\log {2}^{-} 2}$

$= \log \frac{2}{- 2 \log 2}$

$= - \frac{1}{2}$

Hopefully this helps!