ok log properties
[tex]log_ax+log_ay=log_a(xy)[/tex]
and
[tex] \frac{log_ax}{log_ay} log_a(x-y)[/tex]
and
[tex]nlog_ax=log_ax^n[/tex]
so
pemdas
first make like fractions so we can combine
[tex] \frac{2log_bx}{3} + \frac{3log_by}{4} [/tex]
times first fraction by 4/4 and second by 3/3
[tex] \frac{8log_bx}{12} + \frac{9log_by}{12} [/tex]
combine fractions
[tex] \frac{8log_bx+9log_by}{12} [/tex]
now move fractions up
[tex] \frac{log_b(x^8y^9)}{12} [/tex]
now the other part
[tex]\frac{log_b(x^8y^9)}{12}-5log_bz[/tex]
we need to combine that [tex]5log_bz[/tex] with that [tex]\frac{log_b(x^8y^9)}{12}[/tex] by make it als a fraction of common denomenator of 12
multiply [tex]5log_bz[/tex] by 12/12
[tex] \frac{60log_bz}{12} [/tex]
move the coefient to expoment
[tex] \frac{log_bz^{60}}{12} [/tex]
now conbine fractions
[tex]\frac{log_b(x^8y^9)}{12} - \frac{log_b(z^{60})}{12} [/tex]
[tex]\frac{log_b(x^8y^9)-log_b(z^{60})}{12}}{12} [/tex]
apply log property
[tex] \frac{log_b \frac{x^8y^9}{z^{60}} }{12} [/tex]
