SOLUTION: If log(base8) 3=p and log(base3) 5=q, express log(base 10) 5 in terms of p and q.

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Question 253356: If log(base8) 3=p and log(base3) 5=q, express log(base 10) 5 in terms of p and q.
Found 3 solutions by stanbon, Edwin McCravy, Alan3354:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
If log(base8)3=p
8^p = 3
---
and log(base3) 5=q
3^q = 5
---
express log(base10) 5 in terms of p and q.
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Let log(base 10)5 = x
10^x = 5
Then 10^x = 3^q
Then 10^x = (8^p)^q
Then 10^x = 8^(pq)
Then x = log[8^pq]
Then x = (pq)log(base 10)8
----
So, log(base 10)5 = (pq)log(base 10)8
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Cheers,
Stan H.

Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!

We use the change of base formula:

log%28a%2C%28x%29%29=log%28b%2C%28x%29%29%2Flog%28b%2C%28a%29%29

and the basic logarithm formulas.

1.  

therefore  

2.  log%2810%2C%285%29%29=q%2F%28q%2Blog%283%2C%282%29%29%29++



3.  

therefore

    p=1%2F%283log%283%2C%282%29%29%29+

    3p%2Alog%283%2C%282%29%29=1

4.  log%283%2C%282%29%29=1%2F%283p%29

Substitute equation 4 into equation 2

    log%2810%2C%285%29%29=q%2F%28q%2Blog%283%2C%282%29%29%29++
    log%2810%2C%285%29%29=q%2F%28q+%2B1%2F%283p%29%29++
    log%2810%2C%285%29%29=%28q%2F%28q+%2B1%2F%283p%29%29%29%2A%28%283p%29%2F%283p%29%29++
    log%2810%2C%285%29%29=%283pq%29%2F%283pq%2B1%29++ 
 
Edwin

Answer by Alan3354(69443) About Me  (Show Source):