SOLUTION: simplify: log 2+16 log(16/15)+12 log(25/24)+7 log(81/80)

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Question 719127: simplify: log 2+16 log(16/15)+12 log(25/24)+7 log(81/80)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!

First we will combine the logs into one using a property of logs:
log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C%28p%2Aq%29%29
In order to use this property the logs have to have coefficients of 1. So we will start by using another property of logarithms:
n%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5En%29%29
to move the "bad" coefficients into the arguments as exponents:

Now we can combine the logs:
log%28%282%2A%2816%2F15%29%5E16%2A%2825%2F24%29%5E12%2A%2881%2F80%29%5E7%29%29
Now all we have to do is simplify. In order to better see what factors will cancel I am going to start by factoring everything into prime factors:

Simplifying...

log%28%28%282%5E65%2A5%5E24%2A3%5E28%29%2F%283%5E28%2A5%5E23%2A2%5E64%29%29%29
log%28%282%2A5%29%29
log%28%2810%29%29
which is:
1