# How do you condense 2log a + log b – 3log c – log d?

##### 1 Answer
Mar 24, 2016

$\log \left(\frac{{a}^{2} b}{{c}^{3} d}\right)$

#### Explanation:

First, bring the multiplicative constants into the logarithmic expressions using the rule:

$n \log x = \log \left({x}^{n}\right)$

Thus, we obtain:

$= \log \left({a}^{2}\right) + \log b - \log \left({c}^{3}\right) - \log d$

We now use the rule that combines added logarithms through multiplication of their arguments:

$\log m + \log n = \log \left(m n\right)$

Applying this to $\log \left({a}^{2}\right)$ and $\log b$, this yields;

$\log \left({a}^{2} b\right) - \log \left({c}^{3}\right) - \log d$

Opposite to adding, subtracted logarithms can have their arguments divided, in the rule:

$\log m - \log n = \log \left(\frac{m}{n}\right)$

This gives the final answer of

$= \log \left(\frac{{a}^{2} b}{{c}^{3} d}\right)$