Problem 39 - Algebra and Trigonometry

Question

Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \ln 5} & {\text { (b) } \ln 25.3} & {\text { (c) } \ln (1+\sqrt{3})}\end{array} $$

Step-by-Step Solution

Answer

(a) \(\ln 5 \approx 1.6094\); (b) \(\ln 25.3 \approx 3.2317\); (c) \(\ln (1+\sqrt{3}) \approx 1.0040\).

Step 1: Identify the Expression

We need to evaluate three different expressions involving the natural logarithm (\(\ln\)). These are \(\ln 5\), \(\ln 25.3\), and \(\ln (1+\sqrt{3})\).

Step 2: Calculate \(\ln 5\)

Use a scientific calculator to evaluate \(\ln 5\). Enter 5 on the calculator and press the \(\ln\) function key. The result should be approximately 1.6094.

Step 3: Calculate \(\ln 25.3\)

Enter 25.3 into the calculator and press the \(\ln\) function key to find \(\ln 25.3\). The result should be approximately 3.2317.

Step 4: Calculate \(\ln (1+\sqrt{3})\)

First, calculate \(\sqrt{3}\) on the calculator, which is approximately 1.7321. Then, add 1 to get approximately 2.7321. Now, find the natural logarithm of this sum using the \(\ln\) function key. The result should be approximately 1.0040.

Key Concepts

Scientific Calculator

A scientific calculator is a powerful tool that helps in evaluating complex mathematical expressions. It is specially designed for calculations involving logarithms, exponentials, powers, and roots. To effectively use a scientific calculator, it's important to familiarize yourself with its buttons and functions. Here's a simple guide on how to use it:

• Locate the "ln" button for natural logarithms. This button is typically labeled with "ln," and it calculates the natural logarithm of a number.
• For operations like square roots, look for the square root symbol (√) on the calculator.
• Remember, you can enter values directly using the number keys followed by the desired operation key (like "ln" or "√").

Using a scientific calculator not only speeds up the calculation process but also ensures accuracy in decimal places, crucial for tasks requiring precision.

Evaluating Expressions

Evaluating mathematical expressions, especially those involving natural logarithms, is an essential skill in algebra and calculus. To evaluate an expression means to find its numerical value using mathematical rules. Here's how you can approach evaluating expressions:

• Identify the expression you need to evaluate. Make sure to understand each component, such as whether it involves exponentiation, square roots, or other mathematical operations.
• Use your scientific calculator effectively by entering the components in the correct order. For example, in the expression \(\ln (1+\sqrt{3})\), compute the square root first, then add it to 1, and finally compute the logarithm.
• Always round your final answer to the required decimal places, which, in this case, is four decimal places for higher precision.

Practicing the evaluation of different types of expressions helps develop a stronger understanding of mathematical concepts and improves problem-solving skills.

Square Root Calculations

Calculating the square root is a fundamental operation that can be easily performed with a scientific calculator. The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). Square root calculations are often part of larger expressions as seen in expressions like \(\ln (1+\sqrt{3})\).To perform a square root calculation:

• Enter the number you need to find the square root of on the calculator.
• Press the square root button (often depicted as √) to calculate.

The calculator displays the result, which you can use in further computations, such as adding it to another number or applying a logarithm function. Understanding how to calculate and interpret square roots is indispensable for tackling various types of mathematical problems.