# How do you expand ln ((sqrt(a)/bc))?

Oct 17, 2015

$\frac{1}{2} \log \left(a\right) + \log \left(c\right) - \log \left(b\right)$.

#### Explanation:

You could write the argument of the logarithm as $\sqrt{a} \cdot c \cdot \frac{1}{b}$. This is useful because the logarithm of a product is the sum of the logarithms, so

$\log \left(\sqrt{a} \cdot c \cdot \frac{1}{b}\right) = \log \left(\sqrt{a}\right) + \log \left(c\right) + \log \left(\frac{1}{b}\right)$

Now we use the rule which states that $\log \left({a}^{b}\right) = b \log \left(a\right)$. Writing $\sqrt{a}$ as ${a}^{\frac{1}{2}}$, and $\frac{1}{b}$ as ${b}^{- 1}$, we get

$\log \left(\sqrt{a}\right) = \frac{1}{2} \log \left(a\right)$, and $\log \left(\frac{1}{b}\right) = - \log \left(b\right)$.