$$\log_a 0.5 + \log_a 4a - \log_a b - \log_a3b$$
My attempt :
$\log_a (0.5 . 4a) - \log_a \frac{b}{3b} $
$ \log_a \frac{2a x 3b}{b} = \log_a 6a$
Why is my answer wrong ?
According to the book , I'm suppose to get -
$$ \log_a \frac{0.5 x 4a}{b x 3a} $$
Why is there a multiply sign in the denominator ? There is only one '+' sign in the first expression .
Note that $- \log_a b - \log_a3b$ is the same as $$- (\log_a b + \log_a3b)=-\log(b\times 3b)$$ Alternatively, you could do it like this $$ - \log_a b - \log_a3b=\log_a\frac1b+\log_a\frac1{3b} $$ and then continue as you have done.
This is the correct way because that's just how it has been decided that $-$ should work in modern mathematics.
Everything else you've done looks correct to me.
$= log_a (0.5*4a) - (log_a (b*3b)) = log_a (2a) - log_a (3b^2) = log_a ((3b^2)/(2a)) $
