# How do you use the properties of logarithms to rewrite and simplify the logarithmic expression of 2log8^4-1/3log8^8?

May 16, 2015

First you may put all exponents in the log-argument in front of the log:

$= 4 \cdot 2 \log 8 - 8 \cdot \frac{1}{3} \log 8$ then we can combine:

$= \left(8 - \frac{8}{3}\right) \log 8 = \frac{16}{3} \log 8$

If we consider that $8 = {2}^{3}$ we can take this exponent out as well:

$= \frac{16}{3} \log {2}^{3} = 3 \cdot \frac{16}{3} \log 2 = 16 \log 2$

Of course you could do the ${2}^{3}$ thing first. The order of operations does not really matter.