SOLUTION: How do I express this as a single logarithm? log5+ 2log2+log3-log6

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Question 1042383: How do I express this as a single logarithm?
log5+ 2log2+log3-log6
Found 2 solutions by Alan3354, jim_thompson5910:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
How do I express this as a single logarithm?
log5+ 2log2+log3-log6
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Adding logs --> multiply
Subtracting logs ---> division.
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Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming these logs are all base 10.


We'll use the following rules:


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


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


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


Rule 4: log%28b%2C%28b%29%29+=+1 where b is not equal to 1 and b > 0. In this case, I'm assuming the value of b is 10.


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log%28%285%29%29%2B2%2Alog%28%282%29%29%2Blog%28%283%29%29-log%28%286%29%29


log%28%285%29%29%2Blog%28%282%5E2%29%29%2Blog%28%283%29%29-log%28%286%29%29 Use Rule 3


log%28%285%29%29%2Blog%28%284%29%29%2Blog%28%283%29%29-log%28%286%29%29


log%28%285%2A4%29%29%2Blog%28%283%29%29-log%28%286%29%29 Use Rule 1


log%28%2820%29%29%2Blog%28%283%29%29-log%28%286%29%29


log%28%2820%2A3%29%29-log%28%286%29%29 Use Rule 1


log%28%2860%29%29-log%28%286%29%29


log%28%28%2860%29%2F%286%29%29%29 Use Rule 2


log%28%2810%29%29


1 Use Rule 4


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So log%28%285%29%29%2B2%2Alog%28%282%29%29%2Blog%28%283%29%29-log%28%286%29%29 simplifies to 1

In other words, log%28%285%29%29%2B2%2Alog%28%282%29%29%2Blog%28%283%29%29-log%28%286%29%29=1