SOLUTION: how do i solve... logx (cubed root of 9) = 1/6

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Question 211487: how do i solve...
logx (cubed root of 9) = 1/6
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
problem is:
log%28x%2Croot%283%2C9%29%29+=+1%2F6
by the definition of logarithms, this equation is true if and only if:
x%5E%281%2F6%29+=+root%283%2C9%29
that equation is the same as:
x%5E%281%2F6%29+=+9%5E%281%2F3%29
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take the 6th power of both sides of this equation to get:
%28x%5E%281%2F6%29%29%5E6+=+%289%5E%281%2F3%29%29%5E6
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by the laws of exponents, %28x%5Ea%29%5Eb+=+x%5E%28a%2Ab%29 so the equation of %28x%5E%281%2F6%29%29%5E6+=+%289%5E%281%2F3%29%29%5E6 is equivalent to:
x+=+9%5E2
this makes x = 81
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let's see if this holds water.
the original equation is:
log%28x%2Croot%283%2C9%29%29+=+1%2F6
we replace x with 81 to get:
log%2881%2Croot%283%2C9%29%29+=+1%2F6
this is true if and only if 81%5E%281%2F6%29+=+root%283%2C9%29
using the calculator, 81%5E%281%2F6%29 = 2.080083823...
using the calculator root%283%2C9%29 = cube root of 9 = 2.080083823...
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looks like they're equivalent.
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answer is x = 81
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you can also solve this using your calculator by converting from the log to the base of 81, to the log to the base 10 which your calculator can handle.
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log%2881%2Croot%283%2C9%29%29+=+1%2F6
using the logarithm base conversion formula,
log%2881%2Croot%283%2C9%29%29+=+%28log%2810%2Croot%283%2C9%29%29%29%2F%28log%2810%2C81%29%29
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using the calculator,
(log(10,root(3,9)))/(log(10,81))}}} becomes:
.318080836/1.908485019 = .16666666...
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This is equivalent to 1/6 confirming that the answer is correct.
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answer is:
x = 81
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