Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. \(\log _{\frac{1}{2}}(4.7)\)

Step-by-Step Solution

Answer

The expression evaluates to approximately \(-2.23248\).

Step 1: Recall Change-of-Base Formula

The change-of-base formula for logarithms is given by \( \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \), where \(c\) is any positive number different from 1. Commonly, we use natural logs \(\ln\) or common logs \(\log\) for this conversion.

Step 2: Apply Change-of-Base Formula

For the expression \(\log_{\frac{1}{2}}(4.7)\), apply the change-of-base formula using natural logs: \[ \log_{\frac{1}{2}}(4.7) = \frac{\ln(4.7)}{\ln(\frac{1}{2})} \].

Step 3: Evaluate Natural Logs Using a Calculator

Calculate \(\ln(4.7)\) and \(\ln(\frac{1}{2})\) using a calculator. We find that \(\ln(4.7) \approx 1.54756\) and \(\ln(\frac{1}{2}) \approx -0.69315\).

Step 4: Calculate the Quotient

Divide the values obtained: \[ \frac{1.54756}{-0.69315} \approx -2.23248 \].

Step 5: Round to Five Decimal Places

The quotient \(-2.23248\) is already rounded to five decimal places, so no further adjustments are necessary.

Key Concepts

Logarithms

Logarithms are a tool used to determine how many times a specific number, called the base, must be multiplied by itself to achieve another number. In simpler terms, if you have a base "a" and want to know how many times you must multiply it to get "b," the expression is written as \( \log_a(b) \). Logarithms help break complex multiplication into simpler addition problems.

• The term \( \log_a(b) \) is read as “log base a of b.”
• Logs are often used in real-world settings such as measuring the Richter scale for earthquakes or calculating compound interest.

For example, if we need to determine how many times \( \frac{1}{2} \) must be multiplied by itself to reach 4.7, we would express this as \( \log_{\frac{1}{2}}(4.7) \). By using logarithms, this otherwise complex operation becomes a more manageable one.

Natural Logs

Natural logs are a particular type of logarithm where the base is the special mathematical constant \( e \), approximately 2.71828. This kind of logarithm is denoted as \( \ln \), which stands for "logarithmus naturalis." Natural logs are often used in science and engineering due to their link to exponential growth and decay processes.

• They are crucial in situations where growth rates are involved, like calculating continuous compounding in finance.
• The change-of-base formula takes advantage of natural logs for calculating any logarithm with any desired base.

In our exercise, we use \( \ln \) to convert the original expression \( \log_{\frac{1}{2}}(4.7) \) into a more solvable form: \( \frac{\ln(4.7)}{\ln(\frac{1}{2})} \). This conversion is a necessary step in evaluating logs with bases other than \( e \) or 10.

Calculator Usage

Calculators are indispensable for evaluating complex logarithmic expressions, especially those involving natural logs or less common bases. Most scientific calculators come with a specific \( \ln \) button for directly processing natural logarithms.

• First, identify which log value needs computing — for our case, \( \ln(4.7) \) and \( \ln(\frac{1}{2}) \).
• Press the \( \ln \) key, input your number, and obtain your result.

For example, inputting 4.7 yields \( \ln(4.7) \approx 1.54756 \). Calculating \( \ln(\frac{1}{2}) \) gives approximately \(-0.69315\). These values are critical for completing your calculation using the change-of-base formula.

Evaluating Expressions

Evaluating expressions means performing the necessary calculations to solve a given mathematical statement. By using the change-of-base formula, you can transform a log expression into a more straightforward quotient involving natural logs.

• First, compute each natural log independently.
• Next, divide the natural log of the argument by the natural log of the base.

With the calculated \( \ln(4.7) \approx 1.54756 \) and \( \ln(\frac{1}{2}) \approx -0.69315 \), you divide them: \( \frac{1.54756}{-0.69315} \approx -2.23248 \). This quotient represents our original expression, rounded to five decimal places. This process simplifies the task of evaluating expressions that initially seem complex.