# How do you condense log_4(xy)^3-log_4(xy)?

Jul 6, 2018

color(cyan )(=> log (xy)^2

#### Explanation:

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

$\log {x}^{n} = b \log x$

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

$\implies 2 \log \left(x y\right)$

$\implies \log {\left(x y\right)}^{2}$

Jul 6, 2018

$2 \log \left(x y\right) = \log {\left(x y\right)}^{2}$

#### Explanation:

Given: ${\log}_{4} {\left(x y\right)}^{3} - {\log}_{4} \left(x y\right)$

Use the fact that ${\log}_{b} {m}^{n} = n {\log}_{b} m$.

$= 3 {\log}_{4} \left(x y\right) - {\log}_{4} \left(x y\right)$

Let $u = {\log}_{4} \left(x y\right)$.

$= 3 u - u$

$= 2 u$

Substitute back $u = {\log}_{4} \left(x y\right)$ to get:

$= 2 \log \left(x y\right)$

Alternatively, we can bring the exponent back, and it will give us:

$= \log {\left(x y\right)}^{2}$