Problem 36 - 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. \(\log _{4}\left(\frac{15}{2}\right)\)

Step-by-Step Solution

Answer

The value of \( \log_4\left(\frac{15}{2}\right) \) is approximately 1.22925.

Step 1: Understanding the Problem

The given problem is to evaluate the logarithm \( \log_4\left( \frac{15}{2} \right) \) using the change-of-base formula. This means we will convert the logarithm with base 4 into a fraction or quotient of logarithms with a different base, typically the natural logarithm (base \( e \)).

Step 2: Applying the Change of Base Formula

The change-of-base formula allows us to rewrite \( \log_b(x) \) as \( \frac{\log_k(x)}{\log_k(b)} \), where \( k \) is any positive number different from 1, and \( b \) is the base of the original logarithm. Here, we use the natural logarithm, \(\log_{4}\left(\frac{15}{2}\right) = \frac{\ln\left(\frac{15}{2}\right)}{\ln(4)}.\)

Step 3: Calculating the Natural Logarithms

Compute the natural logarithms using a calculator. First, find \( \ln\left(\frac{15}{2}\right) \) and then \( \ln(4) \). You should get approximately: \( \ln\left(\frac{15}{2}\right) \approx 1.70475 \) and \( \ln(4) \approx 1.38629 \).

Step 4: Evaluating the Quotient

Divide the two results obtained from calculating the natural logarithms:\(\frac{\ln\left(\frac{15}{2}\right)}{\ln(4)} = \frac{1.70475}{1.38629}.\) Use the calculator to complete this operation and get approximately 1.22925.

Step 5: Final Result

After evaluating the quotient, we find that the value of \( \log_4\left(\frac{15}{2}\right) \) is approximately 1.22925.

Key Concepts

Natural Logs

Natural logs, also known as natural logarithms, use the mathematical constant \( e \) as their base. This specific logarithm is represented as \( \ln(x) \), where \( e \) is approximately equal to 2.71828. The function provides the power to which \( e \) must be raised in order to achieve a given number. Natural logs are particularly significant in mathematical applications that involve continuous growth or compounding, such as in disciplines like physics, chemistry, and finance.

When you see \( \ln(a) \), it simply means: "to what power must we raise \( e \) to get \( a \)?" For example, \( \ln(e) = 1 \) because \( e^1 = e \). Similarly, \( \ln(1) = 0 \) since any number raised to the power of 0 is 1.

• Remember, \( \ln(e^x) = x \).
• Natural logs can be converted to other bases using the change of base formula.

Logarithms

Logarithms are the inverse operations of exponentiation. They address the question, "to what power must a base be raised to achieve a certain value?" The notation \( \log_b(x) \) is used, where \( b \) is the base, and \( x \) is the number we want to evaluate. For instance, if \( 2^3 = 8 \), then \( \log_2(8) = 3 \).

Logarithms are used to simplify complex calculations, especially when dealing with equations involving exponential growth or decay.

• Common bases include 10 and \( e \), known as common and natural logs, respectively.
• Logarithms have key properties such as \( \log_b(xy) = \log_b(x) + \log_b(y) \).
• The change of base formula is an essential tool for converting logarithms from one base to another.

Calculator Approximation

Calculators are indispensable when it comes to computing logarithms, especially natural logs, to specific decimal places. Technology allows us to find accurate approximations rapidly, which is crucial for practical applications.

In the exercise, we convert the expression \( \log_4\left(\frac{15}{2}\right) \) using natural logs like \( \ln\left(\frac{15}{2}\right) \) and \( \ln(4) \). These evaluations are approximated to five decimal places using a calculator. This method is perfectly accurate for most real-world applications, where tiny discrepancies are acceptable.

• Always use a scientific calculator for these calculations to ensure precision.
• When approximating results, ensure your calculator is set to the correct mode, usually RAD for most logarithms involving \( e \).

Quotient of Logarithms

The quotient of logarithms refers to expressing a logarithm in terms of the division of two other logarithms. This is frequently seen in the change of base formula, which is essential for solving logarithms with bases other than common or natural ones.

The change of base formula is given by \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), where \( k \) can be any positive base (often \( 10 \) or \( e \)). This formula is particularly useful when you need to calculate logarithms with a base not supported directly by calculators.

• This allows the use of calculators to find the logarithm of any base by converting it to a natural log or common log.
• In the exercise, \( \log_4\left(\frac{15}{2}\right) \) is calculated using \( \frac{\ln\left(\frac{15}{2}\right)}{\ln(4)} \).