Simplifying a Logarithmic Expression

Question

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 :).

score 1 · Answer 1 · answered Mar 13 '13 at 15:25

1

Nicely done. ${}{}{}{}{}{}{}{}$

answered Mar 13 '13 at 15:25

Cameron Buie

102,994

So it's correct then? – notverygoodatmaths Mar 13 '13 at 15:26
Yep! ${}{}{}{}$ – Cameron Buie Mar 13 '13 at 15:28
Ah excellent! I've been programming for years, and I've gotten good at that. I've just got gaping holes in my math knowledge. Thanks for that :) – notverygoodatmaths Mar 13 '13 at 15:29

score 1 · Accepted Answer · answered Mar 13 '13 at 15:28

1

Yes you've done that correctly. I like to pull the negative sign up with the exponent to turn everything into addition, but what you've done is fine:

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

answered Mar 13 '13 at 15:28

mv3

1,109

Simplifying a Logarithmic Expression

2 Answers2