Problem 55 - A Graphical Approach to College Algebra

Question

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln 28^{3}$$

Step-by-Step Solution

Answer

\( \ln(28^3) \approx 9.9966 \).

Step 1: Apply the Logarithm Power Rule

We start by applying the power rule of logarithms, which states that \( \ln(a^b) = b \cdot \ln(a) \). Thus, \( \ln(28^3) = 3 \cdot \ln(28) \).

Step 2: Use the Calculator to Find \( \ln(28) \)

Now, use the calculator to find \( \ln(28) \). This gives us approximately \( 3.3322 \).

Step 3: Multiply the Values

Substitute the value of \( \ln(28) \) into the expression from Step 1: \( 3 \cdot 3.3322 = 9.9966 \).

Step 4: Final Result

Thus, the decimal approximation for \( \ln(28^3) \) is approximately \( 9.9966 \).

Key Concepts

logarithm power rule

When working with natural logarithms, it's crucial to understand the logarithm power rule as it simplifies expressions tremendously. This rule states that for any real numbers \(a\) and \(b\), the logarithm of a power can be expressed as the product of the exponent and the logarithm of the base:

• \( \ln(a^b) = b \cdot \ln(a) \)

This formula allows us to break down complex logarithmic expressions into more manageable parts. In the given exercise, \( \ln(28^3) \) is simplified using this rule to become \( 3 \cdot \ln(28) \). By handling the exponent first, the calculation is much easier to perform with or without a calculator.
Understanding this rule not only helps in textbook exercises but also in real-world applications where logarithms play a significant role.

decimal approximation

When you encounter logarithmic functions, especially in natural logarithms, sometimes an exact number isn't feasible. Here, decimal approximation becomes practical. After applying the logarithm power rule, we found that \( \ln(28) \) was needed. Most logarithms do not have neat integer values, so we approximate them using calculators to several decimal places, like \(3.3322\) in this example.

• It bridges the gap between theoretical math and real-world applications.
• Approximations give us workable solutions when exact numbers are either not necessary or too complex to derive.

For students, understanding decimal approximation is key as it aids in effectively communicating and calculating results. Ensuring precision and clarity is vital, especially when these numbers feed into subsequent calculations.

calculator use in math

Using a calculator can significantly expedite solving mathematical exercises, especially those involving complex calculations such as natural logarithms. Calculators simplify finding values like \( \ln(28) \), transforming a potentially complicated computation into a straightforward decimal result.
The key to effective calculator use is knowing how to input expressions correctly and how to interpret the result. Here’s how:

• Ensure your calculator is set to the correct mode (e.g., radians vs. degrees for trigonometric calculations). For logarithms, you usually don't need to switch modes.
• Accurately type the expression to maintain precision. Instead of making multiple approximations, it's better to approximate only once at the final stage.
• Understand the significance of the decimal places presented, particularly in scientific or engineering contexts where precision is crucial.

Being adept at using a calculator not only helps solve problems accurately but also enhances overall numerical literacy.