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Chapter 4: Problem 26

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

Short Answer

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

Step by step solution

01

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.
02

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 \(\sqrt{6}\).
03

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, \(log_{6}(\sqrt{6}) = log_{6}(6^{0.5}) = 0.5\).

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Most popular questions from this chapter

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; 1-\frac{1}{2}+\frac{1}{3} ; 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5} ; \ldots .\) Describe what you observe.

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 3 \ln x+5 \ln y-6 \ln z $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{7}(7 x) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{V x y^{3}}{z^{3}}\right) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{x^{3} y}{z^{2}}\right) $$
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