Question

How do you use the properties of logarithms to expand `log_5 (x^2/(y^2z^3))`?

Answer 1

`2log_5(x) - 2*log_5(y) - 3*log_5(z)`

Explanation:

Property of log #1:

`log_x(a/b) = log_x(a)-log_y(b)`

So: `log_5(x^2/(y^2*z^3))=log_5(x^2)- log_5(y^2*z^3)`

Property of log #2:

`log_x(a*b) = log_x(a)+log_y(b)`

So: `log_5(y^2*z^3) = log_5(y^2) + log_5(z^3)`

Property of log #3:

`log_x (a^n) = n*log_x(a)`

So: `log_5(x^2) = 2log_5(x)`
`log_5(y^2)=2*log_5(y)`
`log_5(z^3) = 3*log_5(z)`

Final Answer:

#log_5(x^2/(y^2*z^3))= 2log_5(x) - [ 2*log_5(y) + 3*log_5(z)] = 2log_5(x) - 2*log_5(y) - 3*log_5(z)#

Hope this helps!

Source: http://www.mathresources.com/products/insidemath/maa/power_property_of_logarithms.html