SOLUTION: Write the expression as a single logarithm with a coefficient of 1. logb(8) + 5[logb(7 + x) − 1/2 logb(6 − x)]

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Question 365314: Write the expression as a single logarithm with a coefficient of 1.
logb(8) + 5[logb(7 + x) − 1/2 logb(6 − x)]
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!

There are three properties of logarithms we will need for this problem:
  • 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
  • q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29

All of these properties can be used in both directions. But for this problem we will using them from left to right. The third property allows us to move a coefficient into the argument as an exponent. We will use it to move the (1/2) in front of the last log into its argument:

Since 1/2 as an exponent means square root, I'll replace the last argument with a square root:

Next, inside the parentheses we have two logarithms of the same base with coefficients of 1 and with a "-" between them. This is a job for the second property:
log%28b%2C+%288%29%29+%2B+5%28log%28b%2C+%28%287+%2B+x%29%2Fsqrt%286+-+x%29%29%29%29
or
log%28b%2C+%288%29%29+%2B+5%2Alog%28b%2C+%28%287+%2B+x%29%2Fsqrt%286+-+x%29%29%29
Next we can use the third property again to move the 5 from in front of the second log:
log%28b%2C+%288%29%29+%2B+log%28b%2C+%28%28%287+%2B+x%29%2Fsqrt%286+-+x%29%29%5E5%29%29
And last, with two logs of the same base with coefficients of 1 and a "+" between them, we can use the first property to combine them:
log%28b%2C+%288%2A%28%287+%2B+x%29%2Fsqrt%286+-+x%29%29%5E5%29%29
which is a single logarithm with a coefficient of 1. (NOTE: You do have a square root in a denominator. So you may need to rationalize the denominator. I'll leave that up to you.)