Problem 56 - Algebra for College Students

Question

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{2 \ln 3}{\ln 8} $$

Step-by-Step Solution

Answer

The result is 1.06.

Step 1: Understand the Formula

The given expression is \( \frac{2 \ln 3}{\ln 8} \), where "\( \ln \)" stands for the natural logarithm, which is the logarithm to the base \( e \). We want to simplify this expression and round the result to the nearest hundredth.

Step 2: Calculate \( \ln 3 \)

Using a calculator or logarithmic tables, calculate \( \ln 3 \). \( \ln 3 \approx 1.0986 \).

Step 3: Calculate \( 2 \times \ln 3 \)

Multiply the natural logarithm of 3 by 2: \( 2 \times 1.0986 = 2.1972 \).

Step 4: Calculate \( \ln 8 \)

Next, calculate \( \ln 8 \). Using a calculator, \( \ln 8 \approx 2.0794 \).

Step 5: Divide the Numerator by the Denominator

Divide the result from Step 3 by the result from Step 4: \( \frac{2.1972}{2.0794} \approx 1.0564 \).

Step 6: Round the Result

Round \( 1.0564 \) to the nearest hundredth. The rounded result is \( 1.06 \).

Key Concepts

Logarithmic Calculations

Understanding logarithmic calculations is crucial in solving the given exercise. A logarithm, specifically the natural logarithm in this case, is a mathematical operation that helps us work with exponential numbers. The natural logarithm, denoted as \( \ln \), uses the base \( e \), which is approximately 2.71828 and arises naturally in many mathematical contexts. In our exercise, we need to calculate \( \ln 3 \) and \( \ln 8 \), which involves finding what power \( e \) must be raised to, in order to equal 3 and 8, respectively. These calculations are often done using a scientific calculator or logarithmic tables due to the non-integer values of \( e \) and its base.

• \( \ln 3 \approx 1.0986 \): This means that \( e^{1.0986} \approx 3 \).
• \( \ln 8 \approx 2.0794 \): Similarly, \( e^{2.0794} \approx 8 \).

Natural logarithms provide a simple way to handle complex exponential calculations and are often used in fields such as physics, biology, and finance.

Mathematical Expressions

The mathematical expression given in the exercise is \( \frac{2 \ln 3}{\ln 8} \). Understanding this expression involves several steps, including recognizing the components and their operations.
First, we have \( 2 \ln 3 \), which requires us to multiply the natural logarithm of 3 by 2. This operation helps amplify the logarithmic value, preparing it for further calculations. Once this multiplication is done, resulting in \( 2.1972 \), we proceed to the division phase.
To simplify \( \frac{2 \ln 3}{\ln 8} \), we divide the computed \( 2 \ln 3 \) by the value of \( \ln 8 \). This division yields a ratio of these two logs, which simplifies the original expression into a more manageable numerical value:

• \( \frac{2.1972}{2.0794} \approx 1.0564 \)

Mathematical expressions like the one we have are designed to solve for precision and clarity in complex calculations, and breaking them down step-by-step ensures a comprehensive understanding of each involved process.

Nearest Hundredth Rounding

Rounding numbers to the nearest hundredth is a key step in finalizing calculations, providing a clearer and simpler answer. In our solution, the calculated value is \( 1.0564 \), and the goal is to round it to the nearest hundredth. The hundredths place is the second digit to the right of the decimal point in a number.
To perform this rounding:

• Examine the third digit (thousandths place) which is 6 in this case.
• Because this digit is 5 or greater, we round the hundredths place up.

Thus, \( 1.0564 \) becomes \( 1.06 \) after rounding. Rounding helps streamline numbers, making them easier to read and use, especially in practical applications where exact precision to many decimal places is unnecessary. Rounding rules are universally applicable and follow straightforward steps, ensuring consistency in mathematical reporting.