Problem 26 - College Algebra

Question

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

Step-by-Step Solution

Answer

The value of \(log_{6}(\sqrt{6}) = 0.5\).

Step 1: Understand the meaning of the square root

Square root of a number is a value that, when multiplied by itself, gives the original number. In this case, \(\sqrt{6}\) is a number which when squared, gives the number 6.

Step 2: Understand the meaning of logarithm

The logarithm is the inverse operation to exponentiation, just as subtraction is the inverse of addition and division is the inverse of multiplication. Logarithms rewrite numbers of the form \(a^{n}\) to \(n \cdot log(a)\). In this case, \(log_{6}(\sqrt{6})\) is seeking the power to which 6 must be raised to obtain \(\sqrt{6}\).

Step 3: Solve the problem

We know from the rules of logarithms that \(log_{a}(a^{n})\) equals \(n\). In this case, since the square root is equivalent to raising a number to the power of 0.5, we can rewrite \(\sqrt{6}\) as \(6^{0.5}\). Thus, \(log_{6}(\sqrt{6}) = log_{6}(6^{0.5}) = 0.5\).