Problem 30 - College Algebra Essentials

Question

Evaluate each expression without using a calculator. $$\log _{6} \sqrt{6}$$

Step-by-Step Solution

Answer

The value of \(\log_{6} \sqrt{6}\) is \(1/2\).

Step 1: Rewrite the square root as an exponent

Begin by recalling that a square root is the same as an exponent of \(1/2\). That allows us to rewrite the problem as \(\log_{6}(6^{1/2})\)

Step 2: Apply the rule of logarithms

The next step is to apply the rule of logarithms that allows us to move an exponent in the argument out in front of the logarithm. This rule states that for any \(a,b > 0\), \(b \neq 1\), and any real numbers \(n\), it holds that \(\log_b (a^n) = n \cdot \log_b(a)\). Applying this rule, we can rewrite the expression as \(1/2 \cdot \log_{6}(6)\)

Step 3: Evaluate the logarithm

Logarithms with the same base and argument return a value of 1. This is due to the definition of the logarithm, which is the inverse operation to exponentiation. So, \(\log_{6}(6) = 1\). Replace \(\log_{6}(6)\) in the expression with \(1\), resulting in \(1/2 \cdot 1 = 1/2\)