SOLUTION: Use the properties of logarithms to write the expression in terms of the logarithms of x, y, and z. logb ( x^3y^2 z^4)^(1/4)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Use the properties of logarithms to write the expression in terms of the logarithms of x, y, and z. logb ( x^3y^2 z^4)^(1/4)       Log On


   

Question 812884: Use the properties of logarithms to write the expression in terms of the logarithms of x, y, and z. logb ( x^3y^2 z^4)^(1/4)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28b%2C+%28%28x%5E3+%2A+y%5E2+%2A+z%5E4%29%5E%281%2F4%29%29%29
There are three properties of logarithms that are often used in problems like this:
  • log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29
  • log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29
  • log%28a%2C+%28p%5En%29%29+=+n%2Alog%28a%2C+%28p%29%29
Since the argument of our log contains just products and powers we will be using just the first and last properties.

First we will use the last property which allows us to move the exponent of the argument out in front:
%281%2F4%29%2Alog%28b%2C+%28x%5E3+%2A+y%5E2+%2A+z%5E4%29%29
The argument is now a product. So we will use the first property to split the log of the product into the sum of the logs of the factors:

Note the parentheses around the logs. It is a good habit to put parentheses around a substitution, especially if there are more terms. The arguments of the three logs are all powers. So we will use the last property again to move the exponents out in front:


Since the expression is now in terms of logs of x, y and z we might be done. But we should probably use the Distributive Property to multiply out the 1/4: