Problem 14 - Intermediate Algebra

Question

Fill in the blanks. 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

Part (a): \(\log_{5}(4x^2 - x) = 1\). Part (b): \(\log_{3}\left(\frac{4x}{7}\right) = 2\).

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

For the equation \( \log _{5} x + \log _{5} (4x - 1) = 1 \), we use the product property of logarithms. This property states that \( \log_b M + \log_b N = \log_b (MN) \). Therefore, we can condense the given expression to a single logarithm:\[ \log _{5} (x(4x - 1)) = 1 \]

Step 2: Final Expression for part (a)

We rewrite the left side of part (a) with the condensed logarithmic expression:\[ \log _{5} (4x^2 - x) = 1 \]Part (a) is now in the form of a single logarithm.

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

For the equation \( \log _{3} 4x - \log _{3} 7 = 2 \), we use the quotient property of logarithms. This property states that \( \log_b M - \log_b N = \log_b \left( \frac{M}{N} \right) \). Therefore, we can condense the given expression to a single logarithm:\[ \log _{3} \left( \frac{4x}{7} \right) = 2 \]

Step 4: Final Expression for part (b)

We rewrite the left side of part (b) with the condensed logarithmic expression:\[ \log _{3} \left( \frac{4x}{7} \right) = 2 \]Part (b) is now in the form of a single logarithm.

Key Concepts

Product Property of Logarithms

The product property of logarithms is a handy tool that streamlines expressions involving the sum of logs. When you see an equation like \( \log_b M + \log_b N \), you can combine these into a single logarithm expression: \( \log_b (MN) \). This property is derived from understanding that logarithms transform multiplication into addition.
For instance, in part (a) of the exercise, we observed this property directly. The equation \( \log _{5} x + \log _{5} (4x - 1) = 1 \) was condensed into a single logarithm: \[ \log _{5} (x(4x - 1)) = 1 \]
Here’s how it worked:

• Identify terms with the same logarithmic base (in this case, 5).
• Recognize the addition between the logs, hinting at multiplication.
• Multiply the expressions inside the logs: \( x \times (4x - 1) = 4x^2 - x \).
• Write them under one log: \( \log_5(4x^2 - x) \).

By using the product property, what might seem like a complex addition of two logs transforms into a neat, single log expression.

Quotient Property of Logarithms

The quotient property of logarithms simplifies expressions involving the difference of logs. When an equation like \( \log_b M - \log_b N \) appears, it can be succinctly expressed as \( \log_b \left( \frac{M}{N} \right) \). This property is based on the logarithmic principle that subtraction mirrors division.
In part (b) of the problem, this was our guiding property. Let’s break it down:

• We started with \( \log _{3} 4x - \log _{3} 7 \).
• The subtraction sign indicates we should divide \( 4x \) by \( 7 \).
• The expression inside the logarithm then becomes \( \frac{4x}{7} \).

This property allows us to represent the problem’s left side as a single logarithmic expression: \[ \log _{3} \left( \frac{4x}{7} \right) = 2 \]
Embracing the quotient property allows for a clean transformation of the logarithmic expressions, reducing complexity and enhancing readability.

Condense Logarithmic Expressions

Condensing logarithmic expressions involves simplifying multiple logarithms into a single expression. This process makes equations easier to manage, providing a clearer view of the problem.
In both parts of the given exercise, we took different approaches to condense the logarithms, employing specific properties based on the structure of the equations.

• In part (a), the product property allowed us to combine terms after recognizing \( \log _{5} x + \log _{5} (4x - 1) = \log _{5} (4x^2 - x) \).
• In part (b), we applied the quotient property to simplify \( \log _{3} 4x - \log _{3} 7 = \log _{3} \left( \frac{4x}{7} \right) \).

The key to condensing an expression lies in correctly utilizing the right property for the operation between the logs – whether it is addition or subtraction.
Remember, every log expression must share the same base for these properties to apply. This condensing technique is particularly valuable as it translates complicated expressions into a more digestible format, making onward calculations or deeper analysis more accessible.