Problem 113 - College Algebra and Calculus: An Applied Approach

Question

The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

Step-by-Step Solution

Answer

The Product Rule in logarithms states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the two numbers, represented as \( \log(ab) = \log(a) + \log(b) \) for arbitrary numbers 'a' and 'b'.

Step 1: Identify the Rule

The Product Rule states that \( \log(ab) = \log(a) + \log(b) \). This rule is fundamental in logarithms.

Step 2: Apply the Rule

We assume 'a' and 'b' as two arbitrary numbers and apply the rule: \( \log(a*b) = \log(a) + \log(b) \). This demonstrates the Product Rule.

Step 3: Validate the Results

To validate the results one can use specific values for 'a' and 'b', and verify if both sides of the equation are equal. For instance if we set 'a' = 2 and 'b' = 3, then \( \log(6) = \log(2) + \log(3) \), both sides of the equation would give the same result, validating that indeed the logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

Key Concepts

Product Rule

The product rule for logarithms is a handy tool that helps us rewrite complex logarithmic expressions. It essentially links multiplication inside a log to addition outside of it. According to this rule, the logarithm of a product is equal to the sum of the logarithms of the multiplicands. In simpler terms, if you have two numbers, say \( a \) and \( b \), the logarithm of their product, \( \log(ab) \), is equal to \( \log(a) + \log(b) \).

This logarithmic property streamlines many algebra problems because it transforms a potentially tricky multiplication problem into a more straightforward addition problem. To put it to the test, try plugging in some numbers for \( a \) and \( b \), like in our example with 2 and 3, where \( \log(6) = \log(2) + \log(3) \). This experimentally confirms the practicality and accuracy of the product rule.

Properties of Logarithms

Logarithms come with a set of properties that make mathematical computations more feasible, especially when dealing with exponential functions. The key properties of logarithms include:

• Product Rule: As discussed, \( \log(ab) = \log(a) + \log(b) \).
• Quotient Rule: This tells us that the logarithm of a quotient is the difference of the logarithms: \( \log\left( \frac{a}{b} \right) = \log(a) - \log(b) \).
• Power Rule: The logarithm of a power involves multiplication: \( \log(a^b) = b \times \log(a) \).

Each of these properties provides a transformative approach to simplifying logarithmic expressions. They allow us to tackle and solve problems that would otherwise be complicated or impossible to compute manually. Understanding these properties is crucial for anyone dealing with exponential growth, finance calculations, and scientific data analyses.

These properties show how logarithms can simplify operations and make them computationally more accessible. They are fundamental concepts taught in math classes and widely used in various real-world applications.

Logarithmic Identities

Logarithmic identities are essential equations that solve for unknowns within logarithms. They form the basis for manipulating and solving logarithmic expressions. Here are some essential logarithmic identities:

• Base Change Formula: This identity helps change one logarithm base to another: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \), where \( c \) can be any positive real number.
• Identity Logarithm: If the argument inside the log is 1, regardless of the base, the result is 0: \( \log_b(1) = 0 \).
• Log of the Base: The logarithm of a base with the same base is 1: \( \log_b(b) = 1 \).

These identities are practical tools for transitioning between different bases and simplifying complex logarithmic expressions.

In mathematics, these identities are invaluable for reducing verbosity and length in problem-solving. They also facilitate easier comparisons and measurements between variables in diverse academic fields. From calculators to coding algorithms, they lay down foundational knowledge for effectively handling logarithms.