# How do you use the properties of logarithms to expand log_10((xy^4)/z^3)?

May 18, 2018

$\implies \log \left(x\right) + 4 \log \left(y\right) - 3 \log \left(z\right)$

#### Explanation:

There are few different properties that we can use here. I have listed them here:

[1] $\log \left(a b\right) = \log \left(a\right) + \log \left(b\right)$

[2] $\log \left(\frac{a}{b}\right) = \log \left(a\right) - \log \left(b\right)$

[3] $\log \left({a}^{b}\right) = b \log \left(a\right)$

We are given:

$\implies \log \left(\frac{x {y}^{4}}{z} ^ 3\right)$

If we apply property [2], we get

$\implies \log \left(x {y}^{4}\right) - \log \left({z}^{3}\right)$

If we apply property [1] to the first term we get

$\implies \log \left(x\right) + \log \left({y}^{4}\right) - \log \left({z}^{3}\right)$

Now we can apply property [3] to all terms to get

$\implies \log \left(x\right) + 4 \log \left(y\right) - 3 \log \left(z\right)$