Problem 114 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The statement is false because it doesn't follow valid logarithmic properties and fails with numerical counterexamples.

Step 1: Understanding the Logarithm Properties

The logarithm of a product property states that \( \log(ab) = \log a + \log b\). Likewise, the logarithm of a quotient is given by \( \log \left(\frac{a}{b}\right) = \log a - \log b\). The key point here is that these properties apply to multiplication and division, not addition.

Step 2: Check the Given Statement

The statement given is \( \log (A + B) = \log A + \log B \). According to the properties of logarithms, this representation is incorrect because it attempts to apply the addition operation within a single logarithm, which is not accounted for by any logarithmic property.

Step 3: Use a Counterexample

To further demonstrate why the statement is false, take some specific numbers, for instance \( A = 2 \) and \( B = 3 \). The left side of the equation becomes \( \log(5) \), while the right side is \( \log(2) + \log(3) \). These two values are not equal because \( \log(5) eq \log(2) + \log(3) \).

Step 4: Conclusion

The attempt to decompose a logarithm of a sum into a sum of logarithms lacks a valid property, and with the counterexample showing that it results in inequality, the statement \( \log (A + B) = \log A + \log B \) is indeed false.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that help simplify complex expressions. These properties only apply to certain operations, namely multiplication and division. Understanding these can clarify where one should use them and where mistakes, like in the given statement, might occur.

Some key properties include:

• Logarithm of a Product: For any positive numbers, the property is expressed as \( \log(a \times b) = \log a + \log b \). This means when you have a product inside a logarithm, it can be divided into the sum of two logarithms.
• Logarithm of a Quotient: Another useful property is \( \log \left(\frac{a}{b}\right) = \log a - \log b \). Here, a quotient inside a logarithm separates into the difference of two logarithms.
• Logarithm of a Power: When dealing with powers, \( \log(a^n) = n \cdot \log a \). This allows the exponent to be brought out in front as a multiplier.

Recognizing and applying these correctly ensures the correct transformation of logarithmic expressions. Notably, none of these apply to addition inside a log, which is a common point of confusion.

Counterexample in Mathematics

A counterexample is a powerful tool to disprove statements in mathematics. When someone claims a general rule or formula is true, providing just one example where it fails is enough to show the statement is false.

In our context, the statement \( \log (A + B) = \log A + \log B \) can be debunked by choosing specific values for \( A \) and \( B \). For instance, if we take \( A = 2 \) and \( B = 3 \), we compute:

• The left side: \( \log(2 + 3) = \log(5) \).
• The right side: \( \log(2) + \log(3) \).

The values do not match, indicating that the equality does not hold. Such a counterexample demonstrates that the property does not exist for logarithms; thus, showcasing the claim's falsehood.

Logarithm of a Product

The property of a logarithm of a product is fundamental in simplifying and solving algebraic expressions involving logs. It states that the logarithm of the product of two numbers equals the sum of the logarithms of those numbers individually.

This is formally written as \( \log(a \times b) = \log a + \log b \). The beauty of this lies in transforming multiplication, often seen as more complex to handle, into addition, which is more straightforward.

Imagine confronting large or unwieldy multiplication within a log; this property efficiently breaks it down. It shines in many algebra and calculus problems, especially when dealing with exponential growth or decay scenarios. Always remember, though, this product rule is distinct from addition, which lacks an equivalent property under the log operation. This distinction can clear up common misunderstandings with the properties of logarithmic functions.