Problem 87 - 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. $$\ln (a+b)+\ln a-\frac{1}{2} \ln 4$$

Step-by-Step Solution

Answer

\( \ln\left(\frac{a^2 + ab}{2}\right) \)

Step 1: Use Logarithm Addition Property

The property of logarithms that deals with addition states that \( \ln x + \ln y = \ln(xy) \). Apply this property to the expression \( \ln(a+b) + \ln a \) to combine them into a single logarithm: \[ \ln((a+b) \cdot a) = \ln(a(a+b)) = \ln(a^2 + ab) \]

Step 2: Use Logarithm Subtraction Property

The subtraction property of logarithms states that \( \ln x - \ln y = \ln\left(\frac{x}{y}\right) \). Apply this property to the expression from Step 1, \( \ln(a^2 + ab) - \frac{1}{2} \ln 4 \). First, express \( \frac{1}{2} \ln 4 \) as \( \ln 4^{1/2} = \ln \sqrt{4} = \ln 2 \). Thus, the expression becomes: \[ \ln\left(\frac{a^2 + ab}{2}\right) \]

Step 3: Simplify the Expression

The given expression has thus been simplified to a form with a single logarithm. Verify that the steps correctly reflect the application of logarithmic properties, resulting in: \[ \ln\left(\frac{a^2 + ab}{2}\right) \]

Step 4: Conclusion

The expression \( \ln(a+b) + \ln a - \frac{1}{2} \ln 4 \) simplifies to \( \ln\left(\frac{a^2 + ab}{2}\right) \) as a single logarithm with coefficient 1.

Key Concepts

Logarithm Addition Property

The logarithm addition property is a handy tool that allows us to combine two separate logarithms into one. This property states that for any positive numbers \(x\) and \(y\), the expression \(\ln x + \ln y\) equals \(\ln(xy)\). It is one of the basic properties of logarithms and helps simplify expressions.

When you are given an expression such as \( \ln(a+b) + \ln a \), you can apply this property to combine them into a single logarithm. For instance, using the property here means you multiply the contents inside the logarithms: \[ \ln((a+b) \cdot a) = \ln(a^2 + ab) \].

By transforming addition into multiplication inside the logarithmic function, the expression begins to streamline. This process is especially useful when you aim to express complex logs in simpler terms.

Logarithm Subtraction Property

Following the idea of simplifying expressions, we have the logarithm subtraction property. This property tells us that \( \ln x - \ln y \) can be rewritten as \( \ln\left(\frac{x}{y}\right) \). Just like addition, this property is crucial in combining and reducing logarithmic terms.

Let's take the expression from our previous simplification, \( \ln(a^2 + ab) \), and subtract another log component, \( \frac{1}{2} \ln 4 \). The fraction \( \frac{1}{2} \) here means we are looking at a square root inside the log, because \( \ln 4^{1/2} = \ln 2 \). This means our subtraction becomes \( \ln(a^2 + ab) - \ln 2 \).

Applying the subtraction property here, you can now divide within the logarithm: \[ \ln\left(\frac{a^2 + ab}{2}\right) \]. This wraps up the subtraction process, leaving the expression neat and compact.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using the properties of logarithms to write them in a reduced, more manageable form. This is essential, especially when dealing with complex equations or in calculus, for further operations.

In our example expression, \( \ln(a+b) + \ln a - \frac{1}{2} \ln 4 \), we've used both addition and subtraction properties of logarithms to bring it down to \( \ln\left(\frac{a^2 + ab}{2}\right) \). Each step strategically reduces the complexity:

• Combining the sums of logs by addition property simplifies the multiplication inside the log.
• Using the subtraction property separates and condenses the logs through division.

Successfully simplifying ensures that the expression is now a single log term with a coefficient of 1, making it much easier to handle in any subsequent mathematical operations or analysis.