Question

evaluate: \\(\log_{27} 9\\)

\\(\bigcirc\\) \\(\frac{1}{3}\\)

\\(\bigcirc\\) \\(\frac{2}{3}\\)

\\(\bigcirc\\) \\(\frac{3}{2}\\)

\\(\bigcirc\\) \\(-\frac{1}{3}\\)

Explanation:

⚡ Using what you learned: Logarithms and Exponential Equations

Step 1: Set up the equation

Set the expression equal to \( x \):

$$ \log_{27}(9) = x $$

Step 2: Rewrite in exponential form

$$ 27^x = 9 $$

Step 3: Express with a common base

Write both sides as powers of \( 3 \):

$$ (3^3)^x = 3^2 $$

$$ 3^{3x} = 3^2 $$

Step 4: Solve for \( x \)

$$ 3x = 2 $$

$$ x = \frac{2}{3} $$

Answer:

$$ \frac{2}{3} $$