Problem 47 - Intermediate Algebra

Question

Explain how to determine \(\log _{7} 46\) without using Property 11.9.

Step-by-Step Solution

Answer

\( \log_7 46 \approx 1.968 \).

Step 1: Recall the Definition of Logarithm

The logarithm, \( \log_b a \), is the exponent to which the base \( b \) must be raised to produce the number \( a \). Here, we want to find \( \log_7 46 \). This means we want to find the power to which 7 must be raised to get 46.

Step 2: Use Logarithm Change of Base Formula

To compute \( \log_7 46 \) without direct computation, use the change of base formula: \( \log_b a = \frac{\log_k a}{\log_k b} \), where \( k \) is any positive number. Commonly, \( k = 10 \) (common logarithm) or \( k = e \) (natural logarithm) is used. Here, we will use the natural logarithm.

Step 3: Apply the Change of Base Calculation

Use the formula with natural logarithms: \( \log_7 46 = \frac{\ln 46}{\ln 7} \). Use a calculator to compute \( \ln 46 \) and \( \ln 7 \).

Step 4: Calculate \( \ln 46 \) and \( \ln 7 \)

Using a scientific calculator, find \( \ln 46 \approx 3.8286 \) and \( \ln 7 \approx 1.9459 \).

Step 5: Divide the Results

Divide the calculated \( \ln 46 \) by \( \ln 7 \): \( \log_7 46 = \frac{3.8286}{1.9459} \approx 1.968 \).

Key Concepts

Change of Base Formula

The change of base formula is a valuable tool when working with logarithms of bases that are not easily calculated. It allows us to express a logarithm in terms of another base, providing us flexibility in calculating values using available resources such as calculators.

The formula is written as:

• \[\log_b a = \frac{\log_k a}{\log_k b}\]

Here, \( b \) is the base of the initial logarithm you want to find, \( a \) is the value, and \( k \) is the base you are changing to.

Typically, people use either base 10 (common logarithm) or base \( e \) (natural logarithm) for \( k \). These bases are chosen because they are commonly available functions in calculators and make the calculations simpler.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special logarithm with base \( e \), where \( e \approx 2.71828 \). It is used extensively in mathematics due to its natural occurrence in growth processes, finance, and calculus.

The natural logarithm is defined as:

• \[\ln x = \log_e x\]

This notation emphasizes the base \( e \). The natural logarithm excels in solving problems involving exponential growth and decay.

When using the change of base formula, the natural logarithm is often chosen due to its integration in scientific calculators and mathematical software, offering convenience and precision.

Scientific Calculator

A scientific calculator is an essential tool for students and professionals working with complex mathematical calculations, including logarithms.

These calculators typically include:

• Functions for common and natural logarithms (\( \log \) and \( \ln \))
• Trigonometric and exponential functions
• Capabilities to handle multiple operations and store memory

To calculate a value like \( \log_7 46 \) using the change of base formula, as shown in the example solution, you would:

• Enter \( \ln 46 \) and \( \ln 7 \) using the natural logarithm function \( \ln \) on your calculator.
• Divide the result of \( \ln 46 \) by \( \ln 7 \).
• Read the final value, which provides \( \log_7 46 \).

This process showcases the capabilities of scientific calculators, making complex logarithmic calculations straightforward.