SOLUTION: Assume that x, y, z, and b are positive numbers and b ≠ 1. Use the properties of logarithms to write the expression as the logarithm of one quantity. −5 logb x &#872

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Assume that x, y, z, and b are positive numbers and b ≠ 1. Use the properties of logarithms to write the expression as the logarithm of one quantity. −5 logb x &#872      Log On


   

Question 1103612: Assume that x, y, z, and b are positive numbers and b ≠ 1. Use the properties of logarithms to write the expression as the logarithm of one quantity.
−5 logb x − 9 logb y + logb z
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

We'll use the following log rules:

Log Rule 1: log%28b%2C%28x%29%29%2Blog%28b%2C%28y%29%29+=+log%28b%2C%28x%2Ay%29%29

Log Rule 2: log%28b%2C%28x%29%29-log%28b%2C%28y%29%29+=+log%28b%2C%28x%2Fy%29%29

Log Rule 3: y%2Alog%28b%2C%28x%29%29+=+log%28b%2C%28x%5Ey%29%29

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-5%2Alog%28b%2C%28x%29%29-9%2Alog%28b%2C%28y%29%29%2Blog%28b%2C%28z%29%29 Start with the given expression

-%285%2Alog%28b%2C%28x%29%29%2B9%2Alog%28b%2C%28y%29%29%29%2Blog%28b%2C%28z%29%29 Factor a negative out of the first two terms

-%28log%28b%2C%28x%5E5%29%29%2Blog%28b%2C%28y%5E9%29%29%29%2Blog%28b%2C%28z%29%29 Use Log Rule 3, shown above, to move the coefficients into exponent positions

-log%28b%2C%28x%5E5%2Ay%5E9%29%29%2Blog%28b%2C%28z%29%29 Use Log Rule 1 to collapse the sum of logs into a log of a product

log%28b%2C%28z%29%29-log%28b%2C%28x%5E5%2Ay%5E9%29%29 Rearrange terms (from the form -m+n to n-m)

log%28b%2C%28z%2F%28x%5E5%2Ay%5E9%29%29%29 Use Log Rule 2 to collapse the difference of logs into a log of a quotient

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So overall, the original expression simplifies to log%28b%2C%28z%2F%28x%5E5%2Ay%5E9%29%29%29