Problem 14 - Elementary and Intermediate Algebra

Question

Use a property of logarithms to condense the left side of each equation to a single logarithm. Do not solve. a. \(\log _{5} x+\log _{5} \cdot(4 x-1)=1\) b. \(\log _{3} 4 x-\log _{3} 7=2\)

Step-by-Step Solution

Answer

a. \(\log_5 (4x^2 - x)=1\); b. \(\log_3 \left(\frac{4x}{7}\right)=2\)

Step 1: Apply the Product Property for part (a)

To condense the expression \( \log_5 x + \log_5 (4x-1) \) into a single logarithm, apply the product property: \( \log_b M + \log_b N = \log_b (M \cdot N) \). Thus, \( \log_5 x + \log_5 (4x-1) = \log_5 (x \cdot (4x-1)) \).

Step 2: Condense part (a)

The expression \( \log_5 (x(4x-1)) \) simplifies to \( \log_5 (4x^2 - x) \), so the condensed form of the left side of the equation is \( \log_5 (4x^2 - x) = 1 \).

Step 3: Apply the Quotient Property for part (b)

For the expression \( \log_3 4x - \log_3 7 \), use the quotient property: \( \log_b M - \log_b N = \log_b \left(\frac{M}{N}\right) \). Therefore, \( \log_3 4x - \log_3 7 = \log_3 \left(\frac{4x}{7}\right) \).

Step 4: Condense part (b)

The expression \( \log_3 \left(\frac{4x}{7}\right) \) represents the condensed form of the left side of the equation \( \log_3 \left(\frac{4x}{7}\right) = 2 \).

Key Concepts

Product Property of Logarithms

The product property of logarithms is a fundamental concept that helps simplify expressions involving the addition of logarithms. It states that the sum of two logarithms with the same base can be expressed as a single logarithm of the product of their arguments. Mathematically, it can be represented as:\[ \log_b M + \log_b N = \log_b (M \cdot N) \]This property is immensely useful when dealing with polynomial and exponential expressions. For example, if you're given an expression like \( \log_5 x + \log_5 (4x-1) \), you can use the product property to combine them into a single logarithm: \[ \log_5 x + \log_5 (4x-1) = \log_5 (x \cdot (4x-1)) = \log_5 (4x^2 - x) \]By doing this, we have managed to condense the expression, making it simpler to work with and aiding in various algebraic manipulations. This property is particularly beneficial in solving logarithmic equations and simplifying complex expressions.

Quotient Property of Logarithms

The quotient property of logarithms allows the subtraction of two logarithms with the same base to be rewritten as a single logarithm whose argument is the quotient of the two original arguments. The formula for this property is:\[ \log_b M - \log_b N = \log_b \left(\frac{M}{N}\right) \]This property is particularly beneficial when trying to simplify expressions or solve logarithmic equations that involve division. Consider the expression \( \log_3 4x - \log_3 7 \). Applying the quotient property, this becomes:\[ \log_3 4x - \log_3 7 = \log_3 \left(\frac{4x}{7}\right) \]Using this property simplifies the expression by condensing it into one logarithm. This simplification can significantly streamline the process of solving equations, as working with one logarithm is typically easier than dealing with multiple separate terms. Understanding how to apply the quotient property is key in handling logarithmic equations efficiently.

Condensing Logarithms

Condensing logarithms is the process of combining several separate logarithmic terms into a single logarithmic expression. This often involves using the product and quotient properties. The process is particularly crucial when trying to solve equations or simplify expressions because it helps reduce complexity and makes further calculations more manageable.Here are a few tips for condensing logarithms:

• Identify terms that can be combined using product or quotient properties.
• Rewrite those terms as one logarithmic expression.
• Make sure the bases of the logarithms are the same before applying properties.

For example, to condense an expression like the one in part (a) of the original exercise, we apply the product property to write:\[ \log_5 x + \log_5 (4x-1) = \log_5 (4x^2 - x) \]And for part (b), we apply the quotient property:\[ \log_3 4x - \log_3 7 = \log_3 \left(\frac{4x}{7}\right) \]Condensing not only simplifies the work but also aids in solving the equations more readily by reducing the number of separate calculations needed.