# How do you calculate Log_2 56 - log_4 49?

Aug 9, 2016

${\log}_{2} 56 - {\log}_{4} 49 = 3$

#### Explanation:

Expression $= {\log}_{2} 56 - {\log}_{4} 49$

First put the each term on the same base. I will choose ${\log}_{2}$

We know that: ${\log}_{b} x = \frac{{\log}_{a} x}{{\log}_{a} b}$

To change the base of ${\log}_{4} 49$ to ${\log}_{2}$
We have $a = 2 , b = 4 \mathmr{and} x = 49$

Hence: ${\log}_{4} 49 = \frac{{\log}_{2} 49}{{\log}_{2} 4}$

$= \frac{{\log}_{2} 49}{2} = {\log}_{2} {49}^{\frac{1}{2}}$

$= {\log}_{2} \sqrt{49} = {\log}_{2} 7$

Therefore, Expression $= {\log}_{2} 56 - {\log}_{2} 7 = {\log}_{2} \left(\frac{56}{7}\right)$
$= {\log}_{2} 8$

$= 3$ (Since $8 = {2}^{3}$)