Problem 5 - College Algebra with Corequisite Support

Question

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\log _{b}\left(\frac{13}{17}\right)\)

Step-by-Step Solution

Answer

\( \log_b\left(\frac{13}{17}\right) = \log_b(13) - \log_b(17) \).

Step 1: Identify the Expression

We expand the logarithmic expression:
For the following exercises, expand each logarithm as much as possible.
Rewrite each expression as a sum, difference, or product of logs.
\(\log _{b}\left(\frac{13}{17}\right)\)

Step 2: Apply Logarithm Rules

• Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Rule: \(\log_b(M/N) = \log_b(M) - \log_b(N)\)
• Power Rule: \(\log_b(M^p) = p\log_b(M)\)

Step 3: Expanded Expression

\( \log_b\left(\frac{13}{17}\right) = \log_b(13) - \log_b(17) \).

Key Concepts

Logarithmic Rules

Logarithmic rules are essential tools that simplify complex logarithmic expressions. They help in breaking down or expanding logarithmic operations into more manageable parts. By understanding these rules, you can transform and manipulate log expressions easily.
Here are some fundamental logarithmic rules:

• Product Rule: The log of a product is the sum of the logs of its factors, expressed as \( \log_b(xy) = \log_b(x) + \log_b(y) \).
• Quotient Rule: The log of a quotient is the difference of the logs of the numerator and the denominator, expressed as \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \).
• Power Rule: The log of a number raised to a power is the power times the log of the base, expressed as \( \log_b(x^y) = y \cdot \log_b(x) \).
• Change of Base Formula: Allows you to change from one base to another, expressed as \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \).

Master these rules to unlock greater flexibility in solving logarithmic problems.

Quotient Rule

The quotient rule is specifically designed for dividing numbers within a logarithmic expression. It simplifies the calculation by converting the division into subtraction. This is helpful because subtraction is often easier to manage, especially with variables.
For an expression like \( \log_b\left(\frac{13}{17}\right) \), applying the quotient rule results in \( \log_b(13) - \log_b(17) \).
This transformation from division into subtraction not only makes the expression simpler but also prepares it for further manipulation, if needed. Understanding how to use the quotient rule effectively allows you to tackle more advanced problems with confidence.

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms. They often appear complex but can be simplified using logarithmic rules. Breaking down these expressions into simpler parts makes them easier to work with.
An expression like \( \log_b\left(\frac{13}{17}\right) \) might initially seem intricate. Yet, using the quotient rule transforms it into the difference of two simpler expressions: \( \log_b(13) - \log_b(17) \).
Simplifying logarithmic expressions is a key skill that allows you to compare, combine, or expand logs effortlessly. Always look for opportunities to apply logarithmic rules to break down expressions into their fundamental components.

Base of a Logarithm

The base of a logarithm is crucial as it dictates the relationship between the numbers involved in the expression. It’s the number which determines how the output value behaves. The base is typically denoted as \( b \) in expressions like \( \log_b(x) \).
Common logarithmic bases include:

• Base 10: Called common logarithm, often written as \( \log(x) \) without a base, is widely used in scientific calculators.
• Base \( e \): Known as natural logarithm, written as \( \ln(x) \), frequently used in calculus and natural sciences.
• Base 2: Important in computer science, written as \( \log_2(x) \), related closely to binary systems.

When working with logs, always check the base as it significantly influences the outcome and understanding of the problem. Different bases have different implications and are applicable in various fields.