Problem 70 - Intermediate Algebra

Question

For Problems \(69-80\), express each of the following as the sum or difference of simpler logarithmic quantities. Assume all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} 5 x $$

Step-by-Step Solution

Answer

\( \log_b(5x) = \log_b 5 + \log_b x \).

Step 1: Apply the Product Rule of Logarithms

Recall that the product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of the factors. Therefore, for the expression \( \log_b(5x) \), we can apply the product rule: \( \log_b(5x) = \log_b 5 + \log_b x \).

Step 2: Simplify the Expression

After applying the product rule in Step 1, the expression \( \log_b(5x) \) is rewritten as \( \log_b 5 + \log_b x \), which is the sum of two simpler logarithmic quantities.

Key Concepts

Product Rule of Logarithms

The product rule is a crucial log property. It helps break down complex expressions into manageable terms. Given a logarithm where two factors are multiplied, like \( \log_b(AB) \), the product rule lets us separate it as \( \log_b A + \log_b B \).

This is handy when simplifying expressions, making it much easier to handle complicated logs. For example, with \( \log_b(5x) \), we see this as a product of two terms: \( 5 \) and \( x \). By the product rule, this turns into \( \log_b 5 + \log_b x \).

• Use this rule when the argument is a product.
• Simplify each factor into logs of individual components.

This step is foundational for breaking down logs into simpler forms.

Properties of Logarithms

Understanding log properties is essential for manipulating and simplifying any logarithmic expression. Besides the product rule, there are several fundamental properties:

• Quotient Rule: The log of a division is the difference of the logs, i.e., \( \log_b \left(\frac{A}{B}\right) = \log_b A - \log_b B \).


• Power Rule: A log with an exponent can be rewritten by moving the exponent to multiply the log: \( \log_b (A^n) = n \log_b A \).


• Change of Base Formula: Allows conversion between different log bases: \( \log_b A = \frac{\log_c A}{\log_c b} \).

These properties are powerful tools to help alter logs into more feasible calculations or to fit the desired form.

Simplifying Logarithmic Expressions

Simplifying logs makes them easier to work with, especially in complex algebra problems. The goal is to turn intimidating logs into straightforward, manageable expressions.

For example, consider simplifying \( \log_b(5x) \). Start by applying the product rule: \( \log_b(5x) = \log_b 5 + \log_b x \). You've now expressed the log in terms of simpler components.

• Break down compounds using product and quotient rules.
• Apply power rules when necessary to deal with exponents.
• Look for opportunities to use log properties for cleaner forms.

By simplifying, you're creating a clear path for solving further problems or evaluating equations involving logs.