Problem 87 - Algebra and Trigonometry

Question

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\)Write each expression in terms of \(A\) and \(C\). $$ \log _{b} \sqrt{\frac{2}{27}} $$

Step-by-Step Solution

Answer

The logarithmic expression \( \log _{b} \sqrt{\frac{2}{27}} \) can be written in terms of \( A \) and \( C \) as \( \frac{1}{2}A - \frac{3}{2}C \)

Step 1: Break down the fraction under the square root

Firstly, \( \sqrt{\frac{2}{27}} \) can be rewritten as \( \sqrt{2} \times \sqrt{\frac{1}{27}} \)

Step 2: Simplify the logarithmic expression

Rewrite \( \sqrt{2} \) and \( \sqrt{\frac{1}{27}} \) in terms of \( b \) using logarithmic properties. Using the rule \( \log_b \sqrt{a}=\frac{1}{2} \log_b a \), the expression becomes \( \frac{1}{2} \log_b 2 - \frac{3}{2} \log_b 3 \)

Step 3: Convert \(\log _{b} 2\) and \(\log _{b} 3\) using given values

Now, substitute \( A \) for \( \log_b 2 \) and \( C \) for \( \log_b 3 \). The final expression becomes \( \frac{1}{2}A - \frac{3}{2}C \)

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve the use of logarithms. A logarithm is essentially the opposite of an exponent. It answers the question: "To what power must we raise the base to get a certain number?" An expression like \( \log_b a \) indicates that you need to find the power that the base \( b \) must be raised to in order to get \( a \).

Understanding logarithmic expressions is crucial because they allow us to explore relationships between numbers through their exponents. When working with complex expressions, like \( \log_b \sqrt{\frac{2}{27}} \), it becomes important to simplify and express them in a form that's easier to work with using known logarithms. In this example, we aim to express it in terms of \( A \) and \( C \), which represent simple logarithms \( \log_b 2 \) and \( \log_b 3 \) respectively.

Break down the components: identify the basics and express new expressions using simpler exponential forms. For example, the square root and fractions involved can be converted into expressions involving known powers.

Logarithmic Properties

Logarithmic properties are the rules and laws that govern how logarithms operate. These properties are essential for simplifying complex logarithmic expressions and solving equations that contain logs. Let's delve into a few important properties that are often handy:

• Product Property: \( \log_b(mn) = \log_b m + \log_b n \)
• Quotient Property: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \)
• Power Property: \( \log_b(m^n) = n \cdot \log_b m \)
• Root Property: \( \log_b \sqrt[n]{m} = \frac{1}{n} \log_b m \)

These properties are behind the foundational transformation of logs. In our original problem, we use the root property to transform \( \log_b \sqrt{\frac{2}{27}} \) into terms of common logs like \( \log_b 2 \) and \( \log_b 3 \). These transformations help in breaking complicated expressions into simpler components for which values (like \( A \) and \( C \)) are already known. Through practice, using these properties becomes intuitive.

Mathematical Substitution

Mathematical substitution is replacing one part of an expression or equation with a new value or expression. It's a valuable technique to simplify expressions and solve equations, especially when dealing with excessively complicated mathematical forms.

In the context of logarithmic expressions, we often have base logarithmic values already known. In the exercise, we're given that \( \log_b 2 = A \) and \( \log_b 3 = C \). These values serve as substitutions for the respective logarithmic components of our expression. When the original expression \( \log_b \sqrt{\frac{2}{27}} \) is broken down using logarithmic properties, the parts \( \log_b 2 \) and \( \log_b 3 \) are substituted by \( A \) and \( C \) respectively, resulting in the expression \( \frac{1}{2}A - \frac{3}{2}C \).

Using substitution not only simplifies the process of handling complex equations but also aids in creating a consistent form that can be used simplifying and comparing different mathematical scenarios. Practice with substitution builds confidence in manipulating logarithms and paves the way for tackling more advanced topics in mathematics.