# simplify this log expression

• Mar 28th 2010, 06:42 PM
differentiate
simplify this log expression

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

is this correct? can I simplify any further?

thanks
• Mar 28th 2010, 06:52 PM
bigwave
$\log_a(MN) = \log_aM+\log_aN
$
• Mar 28th 2010, 06:52 PM
Prove It
Quote:

Originally Posted by differentiate

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

is this correct? can I simplify any further?

thanks

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

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

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

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

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

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

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 $e$.
• Mar 28th 2010, 07:06 PM
differentiate
thanks a bunch Prove it!