Problem 113 - Elementary and Intermediate Algebra

Question

Explain why each statement is false. \(\log A B=(\log A)(\log B)\)

Step-by-Step Solution

Answer

The given statement is false because it misapplies logarithmic rules; the correct rule is \( \log (A \cdot B) = \log A + \log B \).

Step 1: Identify the Incorrect Logarithmic Rule

The statement \( \log A B = (\log A)(\log B) \) is proposing a multiplication rule for logarithms that does not exist. Instead, the correct multiplication rule is \( \log (A \cdot B) = \log A + \log B \).

Step 2: Recall the Correct Logarithmic Properties

To correctly handle expressions involving logarithms, use the properties: \( \log (A \cdot B) = \log A + \log B \) and \( \log \left(\frac{A}{B}\right) = \log A - \log B \). These properties arise from the definition of logarithms as inverse exponents.

Step 3: Why the Given Statement is Incorrect

The statement \( (\log A)(\log B) = \log A B \) implies that multiplying logs is the same as taking the log of a product, which would contradict the addition property of logs. If \( A = 10 \) and \( B = 10 \), then \( \log AB = \log(100) = 2 \), but \( (\log A)(\log B) = (1)(1) = 1 \), not 2.

Key Concepts

Multiplication Rule for Logarithms

When dealing with logarithms, it's crucial to understand their properties and rules. One of the most significant rules is the multiplication rule, which is often misunderstood. This rule doesn't involve multiplying the logarithms themselves. Instead, it tells us how to handle the logarithm of a product.

According to this rule, the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, it is expressed as:

• \( \log(A \cdot B) = \log A + \log B \)

Understanding this concept helps solve logarithmic problems more effectively. This rule is derived from the basic properties of exponents, as logarithms are essentially the inverse of exponential functions. If misunderstood, students might fall into the trap of thinking that \( \log A \cdot \log B \) is correct, which is false.

Logarithm of a Product

The logarithm of a product is one of the fundamental logarithmic properties that learners often encounter first. This property simplifies the process of calculating the logarithm of a complex number or expression.

When you come across \( \log(AB) \), it's important to realize it can be broken down into two simpler parts: \( \log A \) and \( \log B \). Instead of multiplying these two logarithmic values, you simply add them:

• \( \log(AB) = \log A + \log B \)

This approach is handy when dealing with large numbers or cumbersome calculations. Beyond just simplifying calculations, it provides insight into the structure and relationships within mathematics, especially when exploring exponential growth or decay. Misunderstanding this property often leads people to incorrectly multiply the logarithms, which can result in incorrect computations.

Common Logarithmic Mistakes

Learning logarithms can be confusing, especially with all their different properties and rules. A frequent mistake is the misapplication of the properties of logarithms. A common pitfall is misunderstanding the multiplication rule.

Many learners incorrectly assume that multiplying logarithms equals the logarithm of a product, written as \( \log A \cdot \log B = \log(AB) \). This is incorrect because it ignores the fact that in logarithmic operations, the logarithm of a product is actually the sum of the individual logarithms:

• \( \log(AB) = \log A + \log B \)

Another common mistake involves mishandling more complicated expressions, such as thinking there is an existing property for consumption as direct multiplication or division of the logs themselves. Recognizing these mistakes early helps prevent errors in more complex computations and questions, especially those involving algebraic manipulations in equations.