SOLUTION: If log(2/5)=a , then define log(32) in terms of a

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Question 604023: If log(2/5)=a , then define log(32) in terms of a
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
When I looked at this problem, I found a solution by recognizing...
  • Since 32+=+2%5E5 it will be possible to express log(32) in terms of log(2):
    log%28%2832%29%29+=+log%28%282%5E5%29%29+=+5%2Alog%28%282%29%29
    using the property log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29
  • The property log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29 can be used to separate the 2 and the 5:
    log%28%282%2F5%29%29+=+a
    log%28%282%29%29+-+log%28%285%29%29+=+a
    From this we can get an expression for log(2):
    log%28%282%29%29+=+a+%2B+log%28%285%29%29

Substituting this expression into the log(32) expression we get:
log%28%2832%29%29+=+log%28%282%5E5%29%29+=+5%2A%28a+%2B+log%28%285%29%29%29
Distributing the 5 we get:

which expresses log(32) in terms of a.