Problem 57 - Intermediate Algebra

Question

For Problems \(54-61\), perform the following calculations and express answers to the nearest hundredth. (These calculations are in preparation for our work in the next section.) $$ \frac{\ln 5}{2 \ln 3} $$

Step-by-Step Solution

Answer

The answer, rounded to the nearest hundredth, is 0.73.

Step 1: Identify the Expressions

The given expression to evaluate is \( \frac{\ln 5}{2 \ln 3} \). This expression involves the natural logarithm (\( \ln \)) of numbers, 5 and 3, and a division by 2.

Step 2: Calculate Individual Logarithms

First, calculate \( \ln 5 \) and \( \ln 3 \) using a calculator. Make sure to reach precision suitable for further calculation.- \( \ln 5 \approx 1.60944 \) - \( \ln 3 \approx 1.09861 \)

Step 3: Multiply and Simplify the Denominator

Since the denominator has \( 2 \ln 3 \), calculate \( 2 \times \ln 3 \):- \( 2 \times 1.09861 = 2.19722 \).

Step 4: Divide the Logarithms

Now, simplify the expression by dividing \( \ln 5 \) by the result from Step 3:\[ \frac{1.60944}{2.19722} \approx 0.7329 \].

Step 5: Round to the Nearest Hundredth

Finally, round the result from Step 4 to the nearest hundredth:- The value \( 0.7329 \) rounds to \( 0.73 \) when rounded to the nearest hundredth.

Key Concepts

Natural Logarithm

The natural logarithm, often represented as \( \ln \), is a specific logarithm with a base \( e \), where \( e \approx 2.71828 \). It is used frequently in various areas of mathematics and science because \( e \) is a fundamental constant that appears naturally in growth processes, such as population growth or radioactive decay.

When evaluating expressions involving natural logarithms, a scientific calculator is typically used to find the logarithm of a number. Understanding that the natural logarithm translates multiplication into addition and powers into multipliers plays a crucial role in simplifying complex logarithmic expressions.

For example, when you see \( \ln (ab) \), it can be expressed as \( \ln a + \ln b \). Similarly, the expression \( \ln (a^b) \) becomes \( b \cdot \ln a \).

Knowing these properties simplifies many expressions and is especially useful in algebraic manipulation where logarithms are involved.

Expression Evaluation

Expression evaluation is the process of finding the numerical value of a mathematical expression. It involves identifying the components of the expression, performing arithmetic operations, and applying the mathematical principles associated with those operations.

In the given exercise, the expression \( \frac{\ln 5}{2 \ln 3} \) is evaluated. The steps involve:

• Calculating \( \ln 5 \) and \( \ln 3 \) using a calculator to find their approximate numerical values.
• Multiplying \( \ln 3 \) by 2 to complete the operation in the denominator.
• Dividing the result from the top by the result from the bottom to simplify the expression.

Expression evaluation ensures the systematic process of simplifying complex expressions into a single numerical value. Remember, precise calculations at every step are crucial for an accurate final result.

Decimal Approximation

Decimal approximation is an essential tool when dealing with numbers that cannot be expressed exactly as integers or simple fractions. In the context of logarithms and many scientific calculations, the results are often irrational numbers, meaning they cannot be expressed as exact decimal numbers. Therefore, we use approximation methods to express these numbers in decimal form.

In the example \( \frac{\ln 5}{2 \ln 3} \), the result of the calculation was approximated as \( 0.7329 \). Then, it was further rounded to the nearest hundredth to produce \( 0.73 \).

Certain rules are followed while rounding decimals:

• If the number in the thousandths place is 5 or more, increase the number at the hundredth place by one.
• If it is less than 5, leave the number at the hundredth place unchanged.

Rounding facilitates a more manageable representation of numbers and is crucial in ensuring that the result meets the precision required for practical applications.