SOLUTION: Given that log6 base 3=m and log 5 base 6=n. Express log 10 base 3 in terms of m and n

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Question 1145291: Given that log6 base 3=m and log 5 base 6=n. Express log 10 base 3 in terms of m and n
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52817) About Me  (Show Source):
You can put this solution on YOUR website!
.
Since 10 = 2*5, you have 

log%283%2C+%2810%29%29 = log%283%2C+%282%2A5%29%29 = log%283%2C+%282%29%29 + log%283%2C+%285%29%29.    (1)



log%283%2C+%285%29%29 is given to you as the value  n = log%283%2C+%285%29%29.



Your task is to retrieve  log%283%2C+%282%29%29  from the given data.


It is simple:  m = log%283%2C+%286%29%29 = log%283%2C+%282%2A3%29%29 = log%283%2C+%282%29%29 + log%283%2C+%283%29%29 = log%283%2C+%282%29%29 + 1;

hence,  log%283%2C+%282%29%29 = m-1.


Therefore, from the formula (1)  log%283%2C+%2810%29%29 = (m-1) + n = m + n - 1.    ANSWER

Solved.

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Edwin,  thanks for noticing my error !

I was in hurry and mistakenly read  "log 5 base 6"  as   log%283%2C%285%29%29.



Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
I'm sorry, Ikleyn, but the above is incorrect.

    

We substitute 3m-1 for 2 in 2n3n = 5.



We multiply equations 3m-1 = 2 and 3mn= 5.



Edwin