Problem 72 - Algebra 2

Question

Explain why \(\log (5 \cdot 2) \neq \log 5 \cdot \log 2\)

Step-by-Step Solution

Answer

\(\log (5 \cdot 2)\) is equal to \(\log 5 + \log 2\) according to the properties of logarithms. So \(\log (5 \cdot 2) \neq \log 5 \cdot \log 2\).

Step 1: Property of logarithms

Recall the logarithm property: \(\log a(b) = \log a + \log b\). This is to say, the logarithm of the product of two numbers is the sum of the logarithms of those numbers.

Step 2: Applying the properties

If we apply this property to the given expression, \(\log (5 \cdot 2)\), we get \(\log 5 + \log 2\), not \(\log 5 \cdot \log 2\).

Step 3: Misinterpretation of properties

The misinterpretation in the equation \(\log (5 \cdot 2) = \log 5 \cdot \log 2\) arises from confusing multiplication with addition in the context of logarithm properties. Logarithm of the product of numbers equals to the sum of their individual logarithms, not the product.

Key Concepts

Properties of Logarithms

Logarithms come with a set of properties that simplify expressions and make calculations easier. One of the most fundamental properties is how logarithms handle products, quotients, and powers. These properties are essential because they allow you to break down complex expressions into simpler parts.

• Product Property: This states that the logarithm of a product is equal to the sum of the logarithms of the factors. For example, \( \log(ab) = \log a + \log b \).
• Quotient Property: In contrast, the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator, \( \log \left(\frac{a}{b}\right) = \log a - \log b \).
• Power Property: This property allows you to take a power out as a coefficient: \( \log(a^b) = b \cdot \log a \).

These properties are derived from the rules of exponents and are proven using them. Mastering these foundational skills is crucial for solving problems involving logarithms efficiently.

Product Property of Logarithms

The product property of logarithms is a key concept that explains why \( \log(5 \cdot 2) \) is not \( \log 5 \cdot \log 2 \). According to this property, the logarithm of a product is the sum of the logarithms of each factor. This property is grounded in how logarithms convert multiplication into addition.

Consider the expression \( \log (5 \cdot 2) \). By the product property, this simplifies to \( \log 5 + \log 2 \) instead of multiplying the individual logarithms. Thus, the original expression of \( \log(5 \cdot 2) = \log 5 + \log 2 \), and not \( \log 5 \cdot \log 2 \).

Using this property effectively can help when dealing with logarithms in algebra and calculus. It's a tool that transforms multiplication into a straightforward addition, making calculations simpler and less error-prone.

Logarithmic Identities

Logarithmic identities are formulas that relate different logarithmic expressions to one another. These identities are essential in solving logarithmic equations and simplifying complex expressions.

• Change of Base Formula: This identity allows you to change the base of a logarithm. It is expressed as \( \log_b a = \frac{\log_k a}{\log_k b} \), where \( k \) is any positive number.
• Inverse Logarithm: This identity shows that the logarithm and the exponential function are inverses of each other: \( b^{\log_b a} = a \).
• Logarithm of One: Always remember that \( \log_b 1 = 0 \), because any number raised to the power of 0 equals 1.

These identities make logarithms versatile and powerful in handling operations across various mathematical domains. Understanding these and applying them correctly is crucial for students tackling problems involving exponential and logarithmic forms.