# How do you expand log ((3x(5y))/(3z^2))?

Dec 21, 2015

$\log \left(5\right) + \log \left(x\right) + \log \left(y\right) - 2 \log \left(z\right)$

#### Explanation:

Use the following logarithm rules.

$\log \left(A B\right) = \log \left(A\right) + \log \left(B\right)$ Product Rule

$\log \left(\frac{A}{B}\right) = \log A - \log \left(B\right)$ Quotient Rule

$\log \left({A}^{n}\right) = n \log \left(A\right)$ Power Rule

Our problem $\log \left(\frac{3 x \left(5 y\right)}{3 {z}^{2}}\right)$
We can cancel out 3 from numerator and denominator.
$\log \left(\frac{x \left(5 y\right)}{{z}^{2}}\right)$
Then apply the rules which we saw

log((x(5y)) - log(z^2) Applying the Quotient rule.
$\log \left(x\right) + \log \left(5\right) + \log \left(y\right) - \log \left({z}^{2}\right)$ Applying the Product rule.
$\log \left(x\right) + \log \left(5\right) + \log \left(y\right) - 2 \log \left(z\right)$ Applying Power Rule.

Re-arranging.
$\log \left(5\right) + \log \left(x\right) + \log \left(y\right) - 2 \log \left(z\right)$