Problem 66 - Elementary and Intermediate Algebra

Question

Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\log _{b} x^{3} \sqrt{y}\)

Step-by-Step Solution

Answer

\( 3 \cdot \log_b (x) + \frac{1}{2} \cdot \log_b (y) \)

Step 1: Apply the Power Rule

The power rule of logarithms states that for any positive number \(a\), \( \log_b (a^n) = n \cdot \log_b (a) \). Therefore, apply the power rule to the expression \( \log_b (x^3 \sqrt{y}) \) to split the powers, rewriting it as \( \log_b (x^3) + \log_b (y^{1/2}) \).

Step 2: Simplify Using Power Rule

Simplify both terms using the power rule. For \( \log_b (x^3) \), apply the rule to get \( 3 \cdot \log_b (x) \), and for \( \log_b (y^{1/2}) \), apply the rule to get \( \frac{1}{2} \cdot \log_b (y) \).

Step 3: Combine Results

Combine the simplified terms. The expression \( \log_b (x^3 \sqrt{y}) \) simplifies to \( 3 \cdot \log_b (x) + \frac{1}{2} \cdot \log_b (y) \).

Key Concepts

Power Rule of Logarithms

When working with logarithms, understanding the power rule is essential. It helps to break down complex logarithmic expressions into simpler parts. The power rule states that for any positive number \( a \) and any exponent \( n \), the logarithm of \( a \) raised to the power \( n \) is \( n \) times the logarithm of \( a \). Mathematically, it's written as:

• \( \log_b (a^n) = n \cdot \log_b (a) \)

This rule simplifies the computation of logarithms when the argument is a power of another number.
For example, if you have to deal with \( \log_b(x^3 \sqrt{y}) \), you can apply the power rule directly to get:

• \( \log_b(x^3) \) and \( \log_b(y^{1/2}) \)

Breaking down powers within logarithms allows for expressions to be transformed into simpler, more manageable parts.

Logarithm Simplification

Simplifying logarithms involves applying various properties to express a given logarithmic expression in its simplest form. This often involves using the power rule, as well as other logarithmic rules such as the product, quotient, and change of base rules.
The goal is to take a complex expression like \( \log_b(x^3 \sqrt{y}) \) and break it down. By the power rule, this becomes \( 3 \cdot \log_b(x) + \frac{1}{2} \cdot \log_b(y) \).
Now these terms represent the sum of individual logarithms rather than a single, unwieldy expression.

• The expression is more clear and can be easily used in further calculations.
• Removes complexities associated with the original expression by isolating the individual components.

Logarithm simplification is a helpful tool when dealing with equations involving exponential functions or growth rates.

Algebra

Algebra is a branch of mathematics that helps in understanding and manipulating algebraic expressions. When working with logarithms, algebra provides the framework for applying rules and rewriting expressions in different forms.
Using algebraic techniques, you can rewrite logarithms involving powers and roots, making them easier to work with. In our example, algebra allows us to apply the power rule to transform \( \log_b(x^3 \sqrt{y}) \) into a more straightforward format:

• \( 3 \cdot \log_b(x) + \frac{1}{2} \cdot \log_b(y) \).

Additionally, algebraic principles help in combining, expanding, or factoring expressions which is crucial for solving equations or simplifying operations.
This transformation helps streamline complex problems into more workable equations, ultimately aiding in solving real-world problems efficiently.