Question

find the value of the logarithmic expression.
\log_{27} 3

\log_{27} 3 = \square (type an integer or a simplified fraction )

Explanation:

Step1: Recall the logarithm definition

Let \( \log_{27} 3 = x \). By the definition of logarithms, this means \( 27^x = 3 \).

Step2: Express 27 as a power of 3

We know that \( 27 = 3^3 \), so substitute that into the equation: \( (3^3)^x = 3 \).

Step3: Simplify the left side

Using the power of a power rule \( (a^m)^n = a^{mn} \), we get \( 3^{3x} = 3^1 \).

Step4: Set the exponents equal

Since the bases are the same, we can set the exponents equal: \( 3x = 1 \).

Step5: Solve for x

Divide both sides by 3: \( x = \frac{1}{3} \).

Answer:

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