Problem 78 - A Graphical Approach to Precalculus with Limits

Question

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\left(\log _{b} k-\log _{b} m\right)-\log _{b} a$$

Step-by-Step Solution

Answer

\( \log_b \left( \frac{k}{ma} \right) \)

Step 1: Apply the Quotient Property of Logarithms

The expression \( \log_b k - \log_b m \) can be rewritten using the quotient property of logarithms: \( \log_b k - \log_b m = \log_b \left( \frac{k}{m} \right) \). So, the expression becomes \( \log_b \left( \frac{k}{m} \right) - \log_b a \).

Step 2: Apply the Quotient Property Again

Now, consider the entire expression \( \log_b \left( \frac{k}{m} \right) - \log_b a \). Again, use the quotient property of logarithms: \( \log_b \left( \frac{k}{m} \right) - \log_b a = \log_b \left( \frac{\frac{k}{m}}{a} \right) \). Simplifying gives \( \log_b \left( \frac{k}{ma} \right) \).

Step 3: Conclusion: Single Logarithm with Coefficient 1

The expression \( \log_b \left( \frac{k}{ma} \right) \) is the final form of a single logarithm with a coefficient of 1.

Key Concepts

Quotient Property

The quotient property of logarithms is a fundamental concept in understanding how logarithmic expressions can be simplified. This property states that the difference of two logarithms with the same base can be rewritten as a single logarithm of a quotient. For instance, if you have \( \log_b k - \log_b m \), it can be transformed into \( \log_b \left( \frac{k}{m} \right) \). This transformation is valuable because it allows us to condense expressions for easier manipulation.

• This property is essential when combining logarithms.
• It helps when solving equations involving logarithms.

Understanding this property simplifies complex problems and is the first step in rewriting the original expression given in our problem.

Logarithmic Expressions

Logarithmic expressions might seem complex, but they follow some clear principles which make them manageable. A logarithmic expression involves a logarithm, which is an operation that determines the power to which a base number must be raised to produce a given number.

• Logarithms are the inverses of exponentials.
• They are commonly used in various mathematical calculations and real-world applications like measuring sound intensity or pH levels.
• Expressions can often be simplified using properties like product, quotient, and power properties of logarithms.

In our exercise, we focused on transforming these expressions using the logarithmic properties to eventually simplify them into a single logarithm.

Single Logarithm

Rewriting an expression as a single logarithm aims at simplifying the initial expression to its most condensed form with a coefficient of 1. This involves using the properties of logarithms, such as the quotient property, to combine separate logarithmic terms into one.

• This form makes it easier to evaluate or integrate within more extensive mathematical problems.
• It aids in direct comparison or computing without further deconstruction.

By applying the quotient property twice in our provided exercise, we successfully reduced the multi-step logarithmic expression into \( \log_b \left( \frac{k}{ma} \right) \). This process highlights how effective logarithmic properties are in simplifying complex mathematical statements.