Problem 85 - Algebra 2

Question

Evaluate each logarithm. $$ \log _{36} 6 $$

Step-by-Step Solution

Answer

The evaluated logarithm is \(1/2\).

Step 1: Understand the properties of Logarithms

A logarithm \(\log_b a\) can be understood as the power (exponent) to which the base (\(b\)) must be raised in order to obtain \(a\). So in \(\log _{36} 6\), we need to determine what power 36 needs to be raised to, in order to get 6.

Step 2: Simplify the Base

Given our base is 36, we rewrite the base (36) as \(6^2\). This allows us to see how our base and the number we want it to transform into (6) are related.

Step 3: Apply Logarithm Rules

Applying logarithm rules, specifically the power rule which states that \(log_b{a^n} = n \cdot log_b{a}\), we can write \(\log _{36} 6\) as \(\log _{6^2} 6\), which simplifies to \(1/2 \cdot \log _{6} 6\). This follows as we have taken the 2 from the base and brought it in front of the log.

Step 4: Evaluate the Logarithm

When the base and the argument of the log are the same number (\(log_b b\)), the log equals 1. Therefore, \(log _{6} 6\) simplifies to 1, leaving us with \(1/2 \cdot 1 = 1/2 \) as the final solution.