SOLUTION: Determine whether each statement is true or false 3 logb 3√a = logb a

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Question 392818: Determine whether each statement is true or false
3 logb 3√a = logb a
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
3%2Alog%28b%2C+%283sqrt%28a%29%29%29+=+log%28b%2C+%28a%29%29
In order to determine whether this is true or false, we are going to manipulate the left side to see if we can make it look like the right side. But before we start, I am going to replace sqrt%28a%29 with a%5E%28%281%2F2%29%29. (Remember fractional exponents?):
3%2Alog%28b%2C+%283%2Aa%5E%28%281%2F2%29%29%29%29+=+log%28b%2C+%28a%29%29
Since the right side does not have a 3 in front of the logarithm I will start by using the property of logarithms, q%2Alog%28z%2C+%28p%29%29+=+log%28z%2C+%28p%5Eq%29%29, to move the coefficient into the argument as an exponent:
log%28b%2C+%28%283%2Aa%5E%28%281%2F2%29%29%29%5E3%29%29+=+log%28b%2C+%28a%29%29
which simplifies to:
log%28b%2C+%2827%2Aa%5E%28%283%2F2%29%29%29%29+=+log%28b%2C+%28a%29%29
Since 27%2Aa%5E%28%283%2F2%29%29 is clearly not the same as "a" we can see that this (and the original equation) is not a true statement.