# Simplifying a Logarithmic Expression

This is a really, very simple question, but I've never been an extremely confident mathematician and I just want to make sure that my attempt was correct. Oh and this is homework incase you were thinking I'm trying to sneak answers. :) All logarithms are to base b . The original expression is: $$\log(4) - 3\log(1/3) + \log(2)$$

So I decided, the first thing to do was to invoke the power law so they're all in the same form:

$$\log(4) - \log(1/3^3) + \log(2) = \log(4) - \log(1/27) + \log(2)$$

Then I used the subtraction law:

$$\log(4 \cdot 27) + \log(2) = \log(108) + \log(2)$$

And finally I applied the addition law:

$$\log(108 \cdot2) = \log(216)$$

The question was to simplify it to a single logarithm. I just wanted to ensure I had done this right. All the other answers in the paper evaluate to like $log(5)$ and 216 seemed a little out of place :).

$$\log(4) - 3\log(1/3) + \log(2) = \log(4) + \log((1/3)^{-3}) + \log(2)$$
Nicely done. ${}{}{}{}{}{}{}{}$
• Yep! ${}{}{}{}$ – Cameron Buie Mar 13 '13 at 15:28