Question

what is the value of \\(\log_{27} 9\\)?

• \\(-\frac{3}{2}\\)
• \\(-\frac{2}{3}\\)
• \\(\frac{2}{3}\\)
• \\(\frac{3}{2}\\)

Explanation:

⚡ Using what you learned: Evaluating Logarithmic Expressions

Step 1: Set up the equation

Set the expression equal to \( x \):

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

Step 2: Rewrite in exponential form

Convert the logarithmic equation to its exponential form:

$$ 27^x = 9 $$

Step 3: Express with a common base

Write both sides of the equation using the base \( 3 \):

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

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

Step 4: Solve for \( x \)

Equate the exponents:

$$ 3x = 2 $$

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

Answer:

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