# Thread: simplify this log expression

1. ## simplify this log expression

This is what I had:

$\displaystyle \log _354 + log_35 - log_84$
$\displaystyle log_33 + log_33 + log_32 + log_35 - log_8(8^\frac{2}{3})$
$\displaystyle 2\tfrac{1}{3} + log_310$

is this correct? can I simplify any further?

thanks

2. $\displaystyle \log_a(MN) = \log_aM+\log_aN$

3. Originally Posted by differentiate
This is what I had:

$\displaystyle \log _354 + log_35 - log_84$
$\displaystyle log_33 + log_33 + log_32 + log_35 - log_8(8^\frac{2}{3})$
$\displaystyle 2\tfrac{1}{3} + log_310$

is this correct? can I simplify any further?

thanks
$\displaystyle \log_3{54} + \log_3{5} - \log_8{4}$

$\displaystyle = \log_3{(27 \cdot 2)} + \log_3{5} - \log_8{\left(8^{\frac{2}{3}}\right)}$

$\displaystyle = \log_3{27} + \log_3{2} + \log_3{5} - \frac{2}{3}$

$\displaystyle = \log_3(3^3) + \log_3{(2\cdot 5)} - \frac{2}{5}$

$\displaystyle = 3 + \log_3{10} - \frac{2}{5}$

$\displaystyle = \frac{13}{5} + \log_3{10}$.

So yes, you are correct. And no, it can't be simplified any further. The best you could do is to convert it to a base 10 or base $\displaystyle e$.

4. thanks a bunch Prove it!