Problem 1 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{5}(7 \cdot 3) $$

Step-by-Step Solution

Answer

\(\log _{5}(7) + \log _{5}(3)\)

Step 1: Identify the Expression

We expand the logarithmic expression:
In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic
expression as much as possible. Where possible, evaluate logarithmic
expressions without using a calculator.
$$
\log _{5}(7 \cdot 3)
$$

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

\(\log _{5}(7) + \log _{5}(3)\)

Key Concepts

Logarithmic Expressions

Understanding logarithmic expressions is fundamental for any student who wants to master mathematical concepts related to exponents and their inverses. A logarithm is technically the power to which a number, called the base, must be raised to obtain another number. For example, if we have \(\log_b(x)\), it asks the question: ‘To which power do we raise \(b\) to get \(x\)?’ This concept is brilliantly useful in many fields including science, engineering, and finance.

When encountering an expression such as \(\log _{5}(7 \times 3)\), the base here is 5, and we are looking for the power to which 5 must be raised to yield the product of 7 and 3. As intimidating as logarithms might seem at first glance, they are just another way to represent relationships between numbers and have very practical applications, such as in measuring the intensity of sound (decibels) or the acidity of a solution (pH levels).

Moreover, understanding how to work with logarithmic expressions enables students to solve more complicated equations in algebra and calculus, as they commonly appear in various transformations and derivations.

Logarithm Rules

To effectively manipulate and expand logarithmic expressions, it’s crucial to be acquainted with the fundamental rules of logarithms. These rules are straightforward transformation tools that allow us to rewrite logs in different forms for ease of calculation or further simplification. Here's a brief overview of key logarithm rules:

• Product Rule: \(\log_b(m \times n) = \log_b(m) + \log_b(n)\), which tells us that the log of a product can be expressed as the sum of the logs.
• Quotient Rule: \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\), suggesting that the log of a quotient is the difference of the logs.
• Power Rule: \(\log_b(m^n) = n \times \log_b(m)\), meaning that the logarithm of a number raised to a power is that power times the logarithm of the number.
• Base Change Rule: Allows the conversion of a logarithm to a different base through the formula \(\log_b(m) = \frac{\log_k(m)}{\log_k(b)}\), with \(k\) being the new base.
• Logarithm of One: \(\log_b(1) = 0\), because any number raised to the power of zero equals one.
• Logarithm of the base: \(\log_b(b) = 1\), since any number raised to the power of one is the number itself.

Armed with these rules, students can dissect and reconstruct logarithmic expressions with ease, laying the groundwork for more advanced problem-solving.

Expanding Logarithms

Expanding logarithms is a technique used to rewrite complex logarithmic expressions into simpler, more manageable forms. This is often done by applying the logarithm rules such as the product, quotient, and power rules. The aim here is not just to make complex calculations manageable without a calculator, but also to provide a form that enables further algebraic operations.

For instance, the given exercise directed the expansion of \(\log _{5}(7 \times 3)\) by applying the product rule. This rule simplifies the log of a product into a sum of logarithms, thus \(\log _{5}(7 \times 3)\) is expanded to \(\log _{5}(7) + \log _{5}(3)\). It's crucial to understand why this is possible: it's based on the fundamental properties of exponents and how multiplication of numbers translates into addition of their exponents.

Expansion of logarithms not only enhances computational ability but also strengthens the conceptual understanding by connecting algebraic operations with their logarithmic counterparts. This foundational skill is not only useful for solving textbook exercises but is also essential for their real-world applications in analytical domains.