What is the value of x in the equation logx1/81 = 4
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Correct Answer: Option D
Explanation:
To solve the equation \(\log_x \left(\frac{1}{81}\right) = 4\), rewrite it in exponential form:
\[
x^4 = \frac{1}{81}
\]
Since \(\frac{1}{81}\) can be expressed as \(81^{-1}\) and \(81\) is \(3^4\):
\[
\frac{1}{81} = (3^4)^{-1} = 3^{-4}
\]
Thus:
\[
x^4 = 3^{-4}
\]
Taking the fourth root of both sides:
\[
x = 3^{-1} = \frac{1}{3}
\]
So, the value of \(x\) is:
D. \(\frac{1}{3}\)
To solve the equation \(\log_x \left(\frac{1}{81}\right) = 4\), rewrite it in exponential form:
\[
x^4 = \frac{1}{81}
\]
Since \(\frac{1}{81}\) can be expressed as \(81^{-1}\) and \(81\) is \(3^4\):
\[
\frac{1}{81} = (3^4)^{-1} = 3^{-4}
\]
Thus:
\[
x^4 = 3^{-4}
\]
Taking the fourth root of both sides:
\[
x = 3^{-1} = \frac{1}{3}
\]
So, the value of \(x\) is:
D. \(\frac{1}{3}\)