Problem 30 - Introductory and Intermediate Algebra for College Students 4th

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 \( 0.5 \)

Step 1: Identify the problem

The objective is to evaluate \( \log _{6} \sqrt{6} \). In this context, the square root of 6 is equivalent to raising it to the power (exponent) of 0.5.

Step 2: Apply the Logarithm rule

The value of a logarithm is the exponent you have to raise the base to get the argument of the log. In other words, in \( \log _{a} b = c \), \( a^c = b \). Therefore, we must obtain the exponent of the base which gives the argument i.e., \( 0.5 \).

Step 3: Simplify Result

The expression \( \log _{6} \sqrt{6} = \log _{6} 6^{0.5} \) thereby simplifies to \( 0.5 \) because 6 raised to the power of 0.5 equals to square root of 6, which gives us the argument of the log. Hence, the value of the log expression is \( 0.5 \).