SOLUTION: Find the logarithm of log[8](8^(1/7))

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Question 374631: Find the logarithm of log[8](8^(1/7))
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%288%2C+%288%5E%281%2F7%29%29%29
Once you understand what a logarithm is, you will laugh at how easy this problem is!

The general idea of logarithms is that you can take any positive number and, by you raising it to the correct power, get any other positive number. Since many of these exponents are not rational numbers we use special syntax to express these powers.

This is a little like square roots. For example, there is a number you can square to get 2. But it is not a rational number so it is difficult to express. So we use a special syntax, sqrt%282%29, to express this number that you square to get 2.
The special syntax for the exponents is:
log%28a%2C+%28x%29%29
where "a" and "x" must both be positive. This expression represents "the exponent you would put on a to get x. And
log%284%2C+%2820%29%29
represents the exponent you would put on 4 to get 20. IOW: 4%5E%28log%284%2C+%2820%29%29%29+=+20 by definition!

So your expression
log%288%2C+%288%5E%281%2F7%29%29%29
represents the exponent you would put on 8 to get 8%5E%281%2F7%29. Here you can actually see the exponent for 8 that results in 8%5E%281%2F7%29! So
log%288%2C+%288%5E%281%2F7%29%29%29+=+1%2F7!!