SOLUTION: If{{{a^2=b^3=c^5=d^6}}}, then prove {{{ log( d , abc = 31/5 ) }}}

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Question 477461: Ifa%5E2=b%5E3=c%5E5=d%5E6, then prove
+log%28+d+%2C+abc++=+31%2F5+%29+
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
We are given:

a%5E2=b%5E3=c%5E5=d%5E6

Take logs base d: 

log%28d%2Ca%5E2%29=log%28d%2Cb%5E3%29=log%28d%2Cc%5E5%29=log%28d%2Cd%5E6%29

Use the rule of logs that says the log of an exponential is
the exponent times the log of the base of the exponent:

2log%28d%2Ca%29=3log%28d%2Cb%29=5log%28d%2Cc%29=6log%28d%2Cd%29

Use the rule of logs on that last expression that says 
the log of the base of the log is always equal to 1:

2log%28d%2Ca%29=3log%28d%2Cb%29=5log%28d%2Cc%29=6%281%29

2log%28d%2Ca%29=3log%28d%2Cb%29=5log%28d%2Cc%29=6

Set each of the 3 expressions = 6 and solve for the logs:

 2log%28d%2Ca%29=6

Divide both sides by 2

 log%28d%2Ca%29=3

--------------------

 3log%28d%2Cb%29=6

Divide both sides by 3

 log%28d%2Cb%29=2

--------------------

 5log%28d%2Cc%29=6

Divide both sides by 5

 log%28d%2Cc%29=6%2F5

--------------------

Use the rule of logs that says the log of a product 
equals the sum of the logs of the factors: 



Edwin