Problem 37 - College Algebra

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

\(\log_{\frac{1}{2}}(4.7) \approx -2.23258\).

Step 1: Identify the Change-of-Base Formula

The change-of-base formula states that \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \) for any positive number \( c \) different from 1. Here, we will use natural logarithms (base \( e \)).

Step 2: Apply the Change-of-Base Formula

Using the change-of-base formula with natural logarithms, we have \( \log_{\frac{1}{2}}(4.7) = \frac{\ln(4.7)}{\ln(\frac{1}{2})} \).

Step 3: Calculate \( \ln(4.7) \)

Using a calculator, compute the natural logarithm of 4.7. This value is approximately \( \ln(4.7) \approx 1.54756 \).

Step 4: Calculate \( \ln(\frac{1}{2}) \)

Using a calculator, compute the natural logarithm of \( \frac{1}{2} \). This value is approximately \( \ln(\frac{1}{2}) \approx -0.69315 \).

Step 5: Compute the Quotient

Divide the natural logarithm of 4.7 by the natural logarithm of \( \frac{1}{2} \): \[ \frac{1.54756}{-0.69315} \approx -2.23258 \].

Step 6: Round the Result

Round the result to five decimal places: \( -2.23258 \).

Key Concepts

Understanding Logarithms

Logarithms are a mathematical way to express exponents. They answer the question: \( b^x = a \), what is \( x \)? In other words, a logarithm tells us what power we must raise a certain number (the base \( b \)) to obtain another number (\( a \)). For example, \( \log_{10}(100) = 2 \) means \( 10^2 = 100 \).
Logarithms can have different bases, such as 10 (common log), \( e \) (natural log), or any other number. The base is usually written as a subscript. For instance, \( \log_b(a) \) means "log base \( b \) of \( a \)."
When bases are different from 10 or \( e \), we might need to convert using specific rules, such as the change-of-base formula, to simplify calculations.

Natural Logarithms in Detail

Natural logarithms use the number \( e \) as their base, which is approximately 2.71828. We denote natural logs with \( \ln \), so \( \ln(x) \) is the same as \( \log_e(x) \).
Why \( e \)? It's because \( e \) is a fundamental number in mathematics and nature, often appearing in processes that involve growth, decay, and other natural phenomena.

• The function \( e^x \) can be differentiated and integrated very simply, which makes \( e \) very useful.
• Natural logarithms help simplify complex multiplication and division into addition and subtraction, which can be easier to handle, especially when dealing with large numbers.

In calculations, natural logarithms provide a straightforward way to handle exponential growth and decay in fields like science and engineering. This is why problems often use \( \ln \) when applying the change-of-base formula.

Utilizing Calculators for Logarithms

Modern calculators are indispensable for evaluating logarithms, especially when dealing with bases that aren't 10 or \( e \). You might notice your calculator has buttons for \( \log \) (base 10) and \( \ln \) (base \( e \)). Here's how to use them for calculations:

• To find \( \ln(4.7) \), simply input "4.7" and press the \( \ln \) button. This gives \( 1.54756 \).
• For \( \ln(\frac{1}{2}) \), input "0.5" and press the \( \ln \) button. The result will be approximately \( -0.69315 \).

Once you have these values, use them in formulas like the change-of-base formula to find logs with any base. Using a calculator ensures precision, especially when the task involves rounding to several decimal places. They are key tools in efficiently handling complex log operations.

Understanding Base Conversion in Logarithms

Base conversion is crucial when you want to transform logarithms from one base to another. This is where the change-of-base formula shines: \[ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \] Here \( c \) is any positive number not equal to 1.
The formula allows you to convert a logarithm to a more convenient base, such as base 10 or base \( e \), which are easy to input into calculators.
Why do we need this? Sometimes, the base we are working with isn't readily available on our calculators, like \( \log_{\frac{1}{2}}(4.7) \). So, by using a familiar base like \( e \), and finding \( \ln \), it facilitates performing the calculation and ensures accuracy.
Base conversion makes complex logarithmic problems manageable, simplifying the process of evaluating logarithms with unfamiliar bases.