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

Question 211487: how do i solve...
logx (cubed root of 9) = 1/6
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
problem is:

by the definition of logarithms, this equation is true if and only if:

that equation is the same as:

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take the 6th power of both sides of this equation to get:

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by the laws of exponents, so the equation of is equivalent to:

this makes x = 81
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let's see if this holds water.
the original equation is:

we replace x with 81 to get:

this is true if and only if
using the calculator, = 2.080083823...
using the calculator = 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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using the logarithm base conversion formula,

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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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