Problem 47 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to combine the expression. $$\log _{2} A+\log _{2} B-2 \log _{2} C$$

Step-by-Step Solution

Answer

Combine as \( \log_2 \left(\frac{A \cdot B}{C^2}\right) \).

Step 1: Apply the Product Law

The product law of logarithms states that \( \log_b M + \log_b N = \log_b (M \cdot N) \). Using this law, combine \( \log_2 A \) and \( \log_2 B \). This gives us \( \log_2 (A \cdot B) \).

Step 2: Apply the Power Law

The power law of logarithms states that \( n \log_b M = \log_b (M^n) \). Apply this law to \( 2 \log_2 C \). Thus, \( 2 \log_2 C = \log_2 (C^2) \).

Step 3: Apply the Quotient Law

The quotient law of logarithms states that \( \log_b M - \log_b N = \log_b \left(\frac{M}{N}\right) \). Use this law to combine \( \log_2 (A \cdot B) \) and \( \log_2 (C^2) \). This results in \( \log_2 \left(\frac{A \cdot B}{C^2}\right) \).

Key Concepts

Product Law of Logarithms

The Product Law of Logarithms is a key rule that helps simplify logarithmic expressions by leveraging their multiplication properties. In essence, this law states that if you have two logarithms with the same base and you're adding them, you can combine them into a single logarithm. Specifically, if you have \( \log_b M + \log_b N \), you can combine these into \( \log_b (M \cdot N) \). This is possible because logarithms are exponents, and when you add exponents, you effectively multiply their bases.

In practice, this means that if you encounter an expression like \( \log_2 A + \log_2 B \), you can simplify it to \( \log_2 (A \cdot B) \). The key benefit of using the product law is that it reduces the complexity of the expression, allowing for easier analysis or computation. This rule is part of the broader logarithmic rules that help manage expressions by consolidating multiple logarithms into a more manageable form.

Power Law of Logarithms

The Power Law of Logarithms facilitates the manipulation of logarithmic expressions by dealing with coefficients in front of logarithms. It essentially states that multiplying a logarithm by a number is equivalent to raising its argument to the power of that number. In mathematical terms, this is written as \( n \log_b M = \log_b (M^n) \). This transformation is particularly helpful when dealing with expressions where a logarithm is scaled by a constant factor.

For example, if you encounter an expression like \( 2 \log_2 C \), you can rewrite it using the power law as \( \log_2 (C^2) \). This conversion can substantially simplify the expression, as it consolidates the multiplication effect of the coefficient directly onto the argument of the logarithm. This is not only crucial for simplifying expressions but also plays a vital role in solving more complex logarithmic equations, training students to see the inherent connections between multiplication and exponentiation.

Quotient Law of Logarithms

The Quotient Law of Logarithms is one of the foundational laws that assists in simplifying expressions that involve the subtraction of logarithms. According to this law, if you have two logarithms with the same base and you subtract one from the other, you can combine them into a single logarithm representing a division. Specifically, this law states that \( \log_b M - \log_b N = \log_b \left(\frac{M}{N}\right) \).

This means that for an expression like \( \log_2 (A \cdot B) - \log_2 (C^2) \), you can directly use the quotient law to simplify it to \( \log_2 \left(\frac{A \cdot B}{C^2}\right) \). By applying this law, you not only simplify the expression but also demonstrate the power of logarithmic relationships in converting subtractive operations into division. This can be very useful when solving complex logarithmic equations or when trying to reduce expressions to a more comprehensible form for further analysis or computation.