Problem 73 - Algebra for College Students

Question

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} y^{3} z^{4} $$

Step-by-Step Solution

Answer

\( \log_{b} (y^3 z^4) = 3 \log_{b} y + 4 \log_{b} z \)

Step 1: Apply the Product Rule for Logarithms

The product rule for logarithms states that \( \log_{b}(mn) = \log_{b} m + \log_{b} n \). We'll use this property to separate the product inside the logarithm. Apply it to \( \log_{b} (y^3 z^4) \): \[ \log_{b} (y^3 z^4) = \log_{b} (y^3) + \log_{b} (z^4) \]

Step 2: Apply the Power Rule for Each Term

The power rule for logarithms states that \( \log_{b}(a^n) = n \cdot \log_{b}(a) \). Use the power rule for each logarithm term obtained:1. Apply to \( \log_{b} (y^3) \): \[ \log_{b} (y^3) = 3 \cdot \log_{b} y \]2. Apply to \( \log_{b} (z^4) \): \[ \log_{b} (z^4) = 4 \cdot \log_{b} z \]

Step 3: Combine the Results into a Simplified Expression

Combine the terms obtained after applying the power rule into a single expression using addition:\[ \log_{b} (y^3 z^4) = 3 \cdot \log_{b} y + 4 \cdot \log_{b} z \]

Key Concepts

Logarithmic Expressions

In algebra, logarithmic expressions are mathematical statements that involve logarithms, which are operators that help us work with exponential relationships. A logarithm answers the question: "To what power must a certain base be raised, to obtain a given number?" For example, in the expression \( \log_b (x) \), \( b \) is the base, and \( x \) is the number we want to express as a power of \( b \). Logarithms convert multiplication into addition, which can simplify many calculations. This special quality makes logarithms powerful tools in simplifying complex expressions and solving equations.
Understanding how to express logarithmic equations in simpler forms is crucial because it helps to perform calculations more easily and makes the use of logarithms more practical in real-world applications.

Properties of Logarithms

There are several key properties of logarithms that make them versatile in simplifying and computing expressions. Knowing these properties allows us to manipulate and solve logarithmic equations more efficiently:

• **Product Property:** States that the logarithm of a product is the sum of the logarithms of the factors, \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• **Quotient Property:** States that the logarithm of a quotient is the difference of the logarithms, \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \).
• **Power Property:** States that the logarithm of a power is the exponent times the logarithm of the base, \( \log_b(a^n) = n \cdot \log_b(a) \).

Understanding these properties allows us to break down complex logarithmic expressions into simpler components that are easier to work with. This is especially useful in solving exponential equations or evaluating logarithmic statements by hand.

Product Rule for Logarithms

The product rule for logarithms is one of the fundamental properties utilized to simplify logarithmic expressions. It states that
\[ \log_b(mn) = \log_b(m) + \log_b(n) \]
This rule converts the logarithm of a product into a sum of individual logarithms. Its usefulness lies in its ability to simplify expressions involving products of variables or numbers, which are often encountered in algebraic contexts. When faced with a logarithm involving a product, such as \( \log_b(y^3 z^4) \), we apply the product rule to separate it into:
\[ \log_b(y^3) + \log_b(z^4) \]
This step is vital in simplifying the overall expression before applying other logarithmic properties. It highlights how products inside logarithmic expressions can be unraveled into simpler sums.

Power Rule for Logarithms

The power rule for logarithms is another critical property that expands the capability of logarithms to simplify expressions. This rule states that:
\[ \log_b(a^n) = n \cdot \log_b(a) \]
Meaning the power can be taken out front as a coefficient, thereby simplifying the expression. For instance, in the expression \( \log_b(y^3 z^4) \), after applying the product rule, we get \( \log_b(y^3) + \log_b(z^4) \). The power rule allows us to treat these terms as:

• \( \log_b(y^3) = 3 \cdot \log_b(y) \)
• \( \log_b(z^4) = 4 \cdot \log_b(z) \)

These transformations enable the expression to be further simplified to:
\[ 3 \cdot \log_b(y) + 4 \cdot \log_b(z) \]
The power rule's simplicity is valuable when solving logarithmic equations and analyzing exponential functions.