SOLUTION: Write Logb(that is base b)x^3y^2/z(that is a fraction) in terms of the logarithms of x, y, and z.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Write Logb(that is base b)x^3y^2/z(that is a fraction) in terms of the logarithms of x, y, and z.      Log On


   

Question 376797: Write Logb(that is base b)x^3y^2/z(that is a fraction) in terms of the logarithms of x, y, and z.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28b%2C+%28x%5E3y%5E2%2Fz%29%29
There are three properties of logarithms which can be used to manipulate arguments of logarithms:
  • log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29
  • log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29
  • log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29

Since your argument is a fraction/division, we willstart by using the second property to split the numerator and denominator into their own logarithms:
log%28b%2C+%28x%5E3y%5E2%29%29+-+log%28b%2C+%28z%29%29
Next we can use the first property on the first logarithm, since its argument is a product:
log%28b%2C+%28x%5E3%29%29+%2B+log%28b%2C+%28y%5E2%29%29+-+log%28b%2C+%28z%29%29
And last we can use the third property on the first two logarithms, since their arguments have exponents:
3%2Alog%28b%2C+%28x%29%29+%2B+2%2Alog%28b%2C+%28y%29%29+-+log%28b%2C+%28z%29%29
This expression is equivalent to the original expression and it it expressed in terms of the base b logarithms of x, y and z.