SOLUTION: if (125/27)^x = the cubic root of 0.6 then the value of x is _______

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Question 202363: if (125/27)^x = the cubic root of 0.6 then the value of x is _______


Answer by Theo(2978) About Me  (Show Source):
You can put this solution on YOUR website!
your original equation is:
%28125%2F27%29%5Ex+=+%28.6%29%5E%281%2F3%29
by the laws of logarithms, this can only be true if and only if
log%28%28125%2F27%29%2C%28%28.6%29%5E%281%2F3%29%29%29+=+x+
we can convert this log with a base of (125/27) to a log with a base of 10 so that we can solve it using the calculator.
the logarithm base conversion formula is:
log%28%28125%2F27%29%2C%28%28.6%29%5E%281%2F3%29%29%29+=+log%2810%2C%28%28.6%29%5E%281%2F3%29%29%29+%2F+log%2810%2C%28125%2F27%29%29
our formula now becomes:
+log%2810%2C%28%28.6%29%5E%281%2F3%29%29%29+%2F+log%2810%2C%28125%2F27%29%29+ = x.
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now:
log%2810%2C%28%28.6%29%5E%281%2F3%29%29%29 is the same as %281%2F3%29%2A%28log%2810%2C%28.6%29%29%29.
also:
+log%2810%2C%28125%2F27%29%29+ is the same as +log%2810%2C125%29+-+log%2810%2C27%29+
so the equation becomes:
x = %28%281%2F3%29%2A%28log%2810%2C%28.6%29%29%29%29 / +%28log%2810%2C125%29+-+log%2810%2C27%29%29+
you can now use your calculator to solve this.
the equation becomes:
x = %28%281%2F3%29%2A%28-.22184875%29%29%2F%28%282.096910013%29-%281.431363764%29%29
which becomes:
x = %28-0.073949583%29%2F%28.665546249%29
which becomes:
x = -0.1111111111111
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to prove your answer is correct, substitute it in the original equation to get:
%28125%2F27%29%5E%28-0.111111111111%29+=+%28.6%29%5E%281%2F3%29
this becomes:
0.843432665+=+.843432665
which is true.
note that .843432665 = (.6)^(1/3)
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