log(a*b)=log(a)+log(b)
and
logₐbˣ=xlogₐb
also
x^-m=1/(x^m)
and
logₐaˣ=x
so
[tex]log_3 \frac{1}{162} [/tex]=y
[tex]log_3 \frac{1}{2*3^4} [/tex]=y
[tex]log_3 \frac{1}{2} + log_3 \frac{1}{3^4} [/tex]=y
[tex]log_3 \frac{1}{2} + log_3 (3^{-4}) [/tex]=y
[tex]log_3 \frac{1}{2} -4 [/tex]=y
the result is
[tex](log_3 \frac{1}{2})-4 [/tex]
