Problem 57 - Algebra for College Students

Question

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\ln 5}{2 \ln 3} $$

Step-by-Step Solution

Answer

The result is 0.73.

Step 1: Understand the Expression

The expression we need to evaluate is \( \frac{\ln 5}{2 \ln 3} \). Our goal is to simplify and calculate this expression using a calculator for numerical value.

Step 2: Calculate the Individual Logarithms

Using a scientific calculator, find the natural logarithm of 5 and 3 separately. That is, \( \ln 5 \approx 1.6094 \) and \( \ln 3 \approx 1.0986 \).

Step 3: Compute the Denominator

To evaluate \( 2 \ln 3 \), multiply the natural logarithm of 3 by 2: \( 2 \times 1.0986 = 2.1972 \).

Step 4: Evaluate the Fraction

Now divide \( \ln 5 \) by \( 2 \ln 3 \): \( \frac{1.6094}{2.1972} \approx 0.7322 \).

Step 5: Round to Nearest Hundredth

Round the result to the nearest hundredth. The value 0.7322 rounds to 0.73.

Key Concepts

Natural Logarithm

Natural logarithms are a specific type of logarithm with a special base known as Euler's number, typically denoted as "e." This number is approximately equal to 2.71828 and is irrational, meaning its decimal representation goes on indefinitely without repeating. The natural logarithm of a number measures how many times e must be multiplied by itself to reach that number.
Natural logarithms are written as \( \ln x \), where "ln" stands for "natural logarithm" and \( x \) is the number you take the logarithm of. They are particularly useful in scientific calculations because many natural processes, like population growth and radioactive decay, can be described using exponential functions that have "e" as their base.
In the context of the exercise, \( \ln 5 \) and \( \ln 3 \) were calculated to approximate their values when solved numerically using a calculator, arriving at \( \ln 5 \approx 1.6094 \) and \( \ln 3 \approx 1.0986 \).
Understanding the natural logarithm is crucial for diving deeper into calculus and complex scientific calculations.

Scientific Calculations

Scientific calculations often require precision and a thorough understanding of mathematical operations, especially when involving logarithms. These calculations usually utilize a scientific calculator, which is designed to handle a wide range of functions, including logarithms, exponentials, and trigonometric calculations.
In our example, the task required finding the natural logarithm of particular numbers using a scientific calculator to derive accurate values for \( \ln 5 \) and \( \ln 3 \). Calculators simplify these processes by computing the logarithmic values instantly.

Why Are Scientific Calculators Important?

• They save time on complex calculations.
• They offer precision to many decimal places.
• They help verify manual computation outcomes.

Computing expressions such as \( \frac{\ln 5}{2 \ln 3} \), which appear daunting by hand, become manageable with the use of scientific calculators, ensuring that results are concise and rounded to a specified decimal place, like the nearest hundredth for the final result, 0.73, from our solution.

Fraction Simplification

To simplify a fraction, determine whether you can break down the numerator and the denominator, or if you can simplify their division process. This involves understanding the relationship between the components of the fraction and reducing the expression without changing its value.
In the context of logarithmic expressions, simplification sometimes involves computations with approximations to a specific decimal place following arithmetic operations.

Steps to Simplify the Given Expression

• Compute \( 2 \ln 3 \): Multiply \( \ln 3 \) by 2, getting \( 2.1972 \).
• Divide \( \ln 5 \) by the previous result: Calculating \( \frac{1.6094}{2.1972} \) gives approximately \( 0.7322 \).
• Round the result: Since we express to the nearest hundredth, the final answer is 0.73.

The resulting value after simplification is easier to handle and interpret, providing a clear position of the fraction in terms of decimal mathematics, making multi-step scientific evaluations more accessible.