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Chapter 4: Problem 32

Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{\sqrt{3}}$$

Short Answer

Expert verified
-0.5

Step by step solution

01

Understanding the logarithmic expression

Logarithms are ways of expressing numbers, and the expression \(\log _{3} \frac{1}{\sqrt{3}}\) means to what power the base (in this case 3) must be raised to get the number inside the log (in this case \(\frac{1}{\sqrt{3}}\)).
02

Express the number inside the log as exponent of 3

Let's express \(\frac{1}{\sqrt{3}}\) as an exponent of 3. We all know that, \(3^2=9\) and \(3^{-1}=\frac{1}{3}\). So, as per the rules of square roots, \(3^{0.5}=\sqrt{3}\). Thus, \(\frac{1}{\sqrt{3}} = \frac{1}{3^{0.5}} = 3^{-0.5}\). This means our expression can be rewritten as \(\log _{3} {3^{-0.5}}\).
03

Simplify the expression

The general property of logarithm, \(\log_b{b^x} = x\), can be applied now. So, \(\log _{3} {3^{-0.5}} = -0.5\).

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Most popular questions from this chapter

Describe the following property using words: \(\log _{b} b^{x}=x\).

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$\log _{3}(4 x-7)=2$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can solve \(4^{x}=15\) by writing the equation in logarithmic form.

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) At what age is the percentage of some coronary heart disease \(50 \% ?\)

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