Question

Simplify/Condense the attached problem. Thanks!

Answer 1

Ok log properties

$log_ax+log_ay=log_a(xy)$
and
$\frac{log_ax}{log_ay} log_a(x-y)$
and
$nlog_ax=log_ax^n$

so

pemdas
first make like fractions so we can combine
$\frac{2log_bx}{3} + \frac{3log_by}{4}$
times first fraction by 4/4 and second by 3/3
$\frac{8log_bx}{12} + \frac{9log_by}{12}$
combine fractions
$\frac{8log_bx+9log_by}{12}$
now move fractions up
$\frac{log_b(x^8y^9)}{12}$

now the other part
$\frac{log_b(x^8y^9)}{12}-5log_bz$
we need to combine that $5log_bz$ with that $\frac{log_b(x^8y^9)}{12}$ by make it als a fraction of common denomenator of 12
multiply $5log_bz$ by 12/12
$\frac{60log_bz}{12}$
move the coefient to expoment
$\frac{log_bz^{60}}{12}$

now conbine fractions

$\frac{log_b(x^8y^9)}{12} - \frac{log_b(z^{60})}{12}$
$\frac{log_b(x^8y^9)-log_b(z^{60})}{12}}{12}$
apply log property
$\frac{log_b \frac{x^8y^9}{z^{60}} }{12}$