Problem 2 - College Algebra Essentials

Question

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{8}(13 \cdot 7)\)

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \(\log _{8}(13) + \log_{8}(7)\).

Step 1: Understand Logarithm Rules

The properties you need are: the power rule \(\log_b{M^p} = p \log_b{M}\), the product rule \(\log_b{MN} = \log_b{M} + \log_b{N}\), and the quotient rule \(\log_b{\frac{M}{N}} = \log_b{M} - \log_b{N}\). In this case, we only need the product rule as our number inside the logarithm is a product of 13 and 7.

Step 2: Implement the Product Rule

Applying the product rule to the given logarithm \(\log _{8}(13 \cdot 7)\) we get: \(\log _{8}(13) + \log_{8}(7)\)

Step 3: Evaluate If Possible

The expressions \(\log _{8}(13)\) and \(\log_{8}(7)\) cannot be simplified further without a calculator. Therefore, step 2 is our final answer.

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a fundamental tool that helps us to simplify and manipulate logarithmic expressions. When you come across the product of two numbers inside a logarithm, the product rule allows you to transform this product into a sum of logarithms. This property can be mathematically expressed as:\[\log_b{(MN)} = \log_b{M} + \log_b{N}\]In simple terms, the log of a product is equal to the sum of the logs of its factors. This makes it easier to handle and solve equations involving products inside logarithms. For instance, consider the expression \(\log_8(13 \cdot 7)\). Using the product rule, we rewrite it as \(\log_8(13) + \log_8(7)\). This rewriting transforms a more complicated expression into a simpler, more approachable form.

Logarithmic Expressions

Logarithmic expressions are mathematical phrases that involve logarithms, which are the inverse operations of exponentiation. They relate the exponents needed to get from a base number to another number. A basic logarithmic expression is written as \(\log_b{A}\), which reads as "the log base \(b\) of \(A\)."When working with logarithmic expressions, our aim is often to simplify them using various logarithmic rules, such as the product rule, quotient rule, or power rule. These tools help break down complex logarithmic forms into simpler expressions, making calculations or further manipulations much easier to handle. Understanding how to navigate and transform these expressions is a key skill in algebra and calculus.

Expansion of Logarithms

The expansion of logarithms involves using the properties of logarithms to rewrite a single complex logarithmic term into a sum or difference of simpler terms. This process can make analyzing and calculating logarithmic expressions more straightforward.

Why Expand Logarithms?

- Makes it easier to simplify complex logarithmic expressions.- Facilitates solving logarithmic equations by breaking them down into smaller parts.- Helps in identifying values that can be calculated directly without a calculator.

Example: Expanding Using the Product Rule

As in the given problem, the expression \(\log_8(13 \cdot 7)\) can be expanded using the product rule into \(\log_8(13) + \log_8(7)\). This expansion helps in evaluating parts of the expression individually and can make working with larger expressions manageable when solving equations or further simplifying results. Such techniques are valuable for both theoretical understanding and practical applications in mathematics.