Logarithm is the mathematical inverse of exponentiation.
Exponentiation is the mathematical inverse of logarithm.
Best with an example: SQRT(9) which is 3.
Sorry about not being able to mathematically typeset this, but this is a simple editor.
x = SQRT(9)
x = 9 ^ (1/2) since logs are mathematical inverses of exponentiation
take log of both sides
log(x) = log (9^(1/2))
log(x) = (1/2) log (9) now before calculators, you would look up log(9) in a log table
log (x) ~ (1/2) (0.9542)
log (x) ~ 0.47712 since we want x, we do the antilog [log^(1) or 10^x], so back to the table
to find what value yields a log of 0.47712
log^(1) log (x) ~ log^(1) 0.47712
x = 3

The above is for the square root. But this works for all real numbers.
What about the cube root of 10 squared?
Well that's the cube root of 100, which is between 4 and 5, but what about an exact answer.
Trial and error is endless.
x = (10^2) ^ (1/3)
x = 10 ^ (2/3)
log(x) = log [10 ^ (2/3)]
log(x) = (2/3) log(10)
log(x) = (2/3) 1
log(x) ~ 0.6666666...
log^(1) log (x) ~ log^(1) 0.6666666...
x ~ 4.64159
11/27/2013

Edward B.
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