Problem 64 - College Algebra

Question

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{3}\left(\log _{4} x-\log _{4} y\right) $$

Step-by-Step Solution

Answer

The logarithmic expression \( \frac{1}{3}(\log_{4}x - \log_{4}y) \) can be condensed to \( \log_{4}(\sqrt[3]{\frac{x}{y}}) \)

Step 1: Applying Logarithm Properties

The difference of logarithms property states that \( \log_b{A} - \log_b{B} = \log_b {(A/B)} \). Apply this to the expression to combine the two logarithms: \( \frac{1}{3}(\log_{4}x - \log_{4}y) \) becomes \( \frac{1}{3}\log_{4}(\frac{x}{y}) \)

Step 2: Converting the Coefficient to an Exponent

The power rule states that \( a \log_{b} (C) = \log_{b}(C^{a}) \). Apply this property to the transformed logarithmic expression to move the coefficient inside the logarithm. The expression \( \frac{1}{3}\log_{4}(\frac{x}{y}) \) can be written as \( \log_{4}(\sqrt[3]{\frac{x}{y}}) \)

Key Concepts

Logarithmic Expressions

Logarithmic expressions are a way to represent exponents in a concise form. The term "logarithm" comes into play when we need to determine what power we must raise a base number to obtain another number. For example, in the expression \( \log_b{A} \), \( b \) is the base and \( A \) is the result of raising \( b \) to a particular power.
Understanding logarithmic expressions involves recognizing that they are the inverse operations of exponents. Just as multiplication and division are inverses, so are exponentiation and logarithms.
This inverse relationship allows us to manipulate and simplify complex mathematical equations efficiently, turning products into sums or differences and powers into simple multipliers using specific properties of logarithms.

• Expression Example: \( \log_2{8} = 3 \) because \( 2^3 = 8 \).
• Base \( b \) must always be a positive number, and not equal to one.
• Expressions can be condensed or expanded using logarithmic properties.

Difference of Logarithms

The property known as the "difference of logarithms" helps us simplify expressions that involve the subtraction of two logarithms. This property is expressed as \( \log_b{A} - \log_b{B} = \log_b {\left(\frac{A}{B}\right)} \).
Using this property, you can consolidate two separate logarithms into a single one. This is immensely helpful while solving equations as it allows us to see the relationship between the numbers more clearly.
Essentially, when you have the same base in a subtraction of logs, you can convert it into a division inside a single log. Here's how this was applied in the original problem:

• The expression \( \log_4{x} - \log_4{y} \) converts to \( \log_4{\left(\frac{x}{y}\right)} \).
• Both terms need to have the same base for this property to be used effectively.

The usage of this property simplifies complex expressions and equations, allowing us to condense them more effectively.

Power Rule for Logarithms

The power rule for logarithms provides a way to manipulate logs when they involve a coefficient multiplied by the logarithm across. It states that \( a \log_b{C} = \log_b{(C^a)} \), meaning you can move the multiplier as an exponent of the base inside the log.
This rule is beneficial, especially when you're dealing with expressions where the log is being multiplied by a fraction or another constant, like in our exercise. This transformation allows us to condense and better manage logarithmic expressions.
In our exercise, the property was used in this way:

• The expression \( \frac{1}{3} \log_4{\left(\frac{x}{y}\right)} \) transforms into \( \log_4{\left(\left(\frac{x}{y}\right)^{1/3}\right)} \).
• The multiplier \( \frac{1}{3} \) acts as an exponent inside the log, creating a cube root.

This rule is one of the fundamental properties of logarithms, commonly used in both computational and theoretical mathematics.