Problem 32 - Algebra and Trigonometry

Question

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{3} \frac{1}{\sqrt{3}} $$

Step-by-Step Solution

Answer

\(\log_{3} \frac{1}{\sqrt{3}} = -1/2\)

Step 1: Rewrite the Argument in Terms of Base

Rewrite \(1/\sqrt{3}\) as \(3^{-1/2}\). This is because \(\sqrt{a}\) can be rewritten as \(a^{1/2}\), and 1/a is \(a^{-1}\). So, \(1/\sqrt{3}\) can be rewritten as \(3^{-(1/2)}\). Hence, the given expression becomes \(\log_{3} 3^{-(1/2)}\).

Step 2: Simplify the Logarithm

Use the property of logarithm that states \(\log_{b} b^{a}=a\). Hence, \(\log_{3} 3^{-1/2}=-1/2\).

Step 3: Write the Final Result

So, \(\log_{3} \frac{1}{\sqrt{3}} = -1/2\)

Key Concepts

Logarithmic Properties

When delving into logarithms, it's essential to understand their fundamental properties. A logarithm answers the question: "To what exponent must I raise the base to obtain another number?" In this exercise, we are working with the expression \( \log_{3} \frac{1}{\sqrt{3}} \). Logarithmic properties can often simplify complex expressions, making them easier to evaluate.

One key property of logarithms, which we used in the solution, is the power rule:

• \( \log_{b}(b^a) = a \)

This property indicates that if you have a number raised to an exponent and you are taking the logarithm of that number with the base being the same as the number's base, the result is simply the exponent. This was crucial in our problem, allowing us to directly simplify and evaluate the logarithm.

Exponent Rules

Exponents are closely connected to logarithms because they are essentially opposite operations. In the problem \( \log_{3} \frac{1}{\sqrt{3}} \), we used exponent rules to reframe the expression. Converting the expression involves understanding several key exponent rules:

• The square root of a number \( a \) is represented as \( a^{1/2} \).
• Reciprocals of numbers use negative exponents: \( \frac{1}{a} = a^{-1} \).

Using these rules, \( \frac{1}{\sqrt{3}} \) becomes \( 3^{-1/2} \). This simplification is vital because it allows us to apply logarithmic properties easily in the next steps. Being comfortable with these exponent rules can help streamline solving logarithmic equations and expressions in general.

Logarithmic Expressions

Logarithmic expressions might look intimidating at first glance, but knowing how to manipulate them with the properties of logs and exponents can greatly simplify these expressions. For the expression \( \log_{3} \frac{1}{\sqrt{3}} \), recognizing the need to adjust its form was key. Here's what we did:

By rewriting the expression \( \frac{1}{\sqrt{3}} \) as \( 3^{-1/2} \), we linked it directly with the base of the log, which is 3. This association is vital as it connects the core of the logarithmic expression with the base, simplifying the calculation.

Once this connection is made, the conversion of the expression into a more manageable form allows the direct application of logarithmic properties, as seen in our solution. Understanding how to adjust and evaluate logarithmic expressions by exploiting these aspects can assist in deconstructing and solving more complicated logarithmic problems.