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Chapter 4: Problem 30

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

Short Answer

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

Step by step solution

01

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 \( \log_{b} b = 1 \).
02

Recognize the square root

The square root of a number \(x\) is the same as \(x^{1/2} \). In this case, \( \sqrt{6} = 6^{1/2} \).
03

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 \( \log_{6} 6^{1/2} = 1/2 \).

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Most popular questions from this chapter

Write as a single term that does not contain a logarithm: \(e^{\ln 8 x^{3}-\ln 2 x^{2}}\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\log _{2} 8}{\log _{2} 4}=\frac{8}{4}$$

Evaluate or simplify each expression without using a calculator. $$ e^{\ln 5 x^{2}} $$

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40\).

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