Question

use the elementary properties of logarithms to solve the following equation. write your answer as a fraction reduced to lowest terms.\\(\log_{27}(x) = \frac{2}{3}\\)\
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\\(x = \\)\
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Explanation:

Step1: Recall log definition

The logarithmic equation $\log_{b}(a) = c$ is equivalent to $b^c = a$.
Here, $b = 27$, $c=\frac{2}{3}$, so we rewrite the equation as $27^{\frac{2}{3}}=x$.

Step2: Simplify the exponent

We know that $27 = 3^3$. Substitute this into the equation: $(3^3)^{\frac{2}{3}}=x$.
Using the exponent rule $(a^m)^n=a^{m\times n}$, we get $3^{3\times\frac{2}{3}} = x$.
Simplify the exponent: $3^2=x$.

Step3: Calculate the result

$3^2 = 9$, so $x = 9$.

Answer:

$x = \boxed{9}$