Problem 30 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the properties of logarithms

In particular, the change of base formula might be helpful. The change of base formula is \( \log_{b} a = \frac{\log_{k} a}{ \log_{k} b} \), where \(b\), \(a\) and \(k\) are all positive real numbers and \(b\neq 1, k\neq 1) \. In this case here, however, it's possible that simpler logarithm laws may be used instead, such as the rule that \( \log_{b} b = 1 \).

Step 2: Recognize the square root

The square root of a number \(x\) is the same as \(x^{1/2} \). In this case, \( \sqrt{6} = 6^{1/2} \).

Step 3: Apply logarithm laws

Consider that a logarithm of a number to a certain base is asking 'To what exponent must the base be raised to get the number?'. 'log_6 6^(1/2)' is asking '6 raised to what power equals 6^1/2?'. It is clear that the power is 1/2, so \( \log_{6} 6^{1/2} = 1/2 \).