How do you expand this logarithm?

log_(3)(z^(4)sqrtx)log3(z4x)

2 Answers
Nov 21, 2016

4log_3 (z) + 1/2log_3(x)4log3(z)+12log3(x)

Explanation:

General Rules:
color(white)("XXX")log_b a^c = c * log_b aXXXlogbac=clogba

color(white)("XXX")log_b (a * c) = log_b + log_b cXXXlogb(ac)=logb+logbc

Nov 21, 2016

4log_3(z)+1/2log_3(x)4log3(z)+12log3(x).

Explanation:

By using the rule of logarithms where log(a*b)=loga+logblog(ab)=loga+logb, we first get

log_3(z^4sqrtx)log3(z4x)
=log_3(z^4)+log_3(sqrtx)=log3(z4)+log3(x)
=log_3(z^4)+log_3(x^(1//2))=log3(z4)+log3(x1/2)

Another rule of logarithms is log(a^b)=blog(a)log(ab)=blog(a). We now use this to get

=4log_3(z)+1/2log_3(x)=4log3(z)+12log3(x)

Unless you have been asked to rewrite the "base 3" logarithms in "base 10" form, this is as much expansion as we can do.

Hope this helps!