Problem 53 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the logarithm. Round your result to three decimal places.\(\ln \sqrt{6}\)

Step-by-Step Solution

Answer

The natural logarithm of the square root of 6, rounded to three decimal places, is approximately 0.895.

Step 1: Transform the expression

Translate the original expression \(\ln \sqrt{6}\) using the logarithmic property that \(\ln \sqrt{x}\) is equal to \(0.5 \cdot \ln{x}\). This gives us \(0.5 \cdot \ln{6}\).

Step 2: Calculate the natural logarithm

Evaluate \(\ln{6}\) using a calculator.

Step 3: Multiply by 0.5

Multiply the result from the last step by 0.5 to get the final answer.

Key Concepts

Logarithmic Properties

To solve the exercise in a more straightforward manner, we need to understand the properties of logarithms, which make calculations more manageable. One key property is the power rule of logarithms. This rule states that the logarithm of a square root can be represented as half of the logarithm of the number itself. In mathematical terms, this means:

• \( \ln \sqrt{x} = 0.5 \times \ln x \)

By applying this property, we can transform more complex expressions into simpler ones, like turning \( \ln \sqrt{6} \) into \( 0.5 \times \ln 6 \). Understanding these properties allows us to manipulate and solve logarithmic expressions that might otherwise seem quite challenging.
Therefore, whenever you encounter a logarithm of a power or a root, remember that these logarithmic properties can simplify your calculations significantly. They are essential tools for managing logarithms efficiently.

Calculator Usage

To evaluate logarithms, such as \( \ln 6 \), efficiently and accurately, we use a scientific calculator. Most calculators offer straightforward functionality to compute natural logarithms (denoted \( \ln \)). Here’s how you can do it:

• First, turn on your calculator and look for the 'ln' button. This button is typically marked with 'ln' for natural logarithm.
• Input the number for which you need to calculate the logarithm, in this case, 6.
• Press the 'ln' button. The display will show the value of \( \ln 6 \) immediately.

After obtaining the initial logarithm value, follow any steps needed to complete our calculation, like multiplying by 0.5 in this task. Using a calculator makes this process smooth and error-free, especially when dealing with more complex or non-integer values. It's important to note that different calculators might have slightly different interfaces, so getting familiar with yours will always be beneficial.

Decimal Rounding

In mathematics, precision is often crucial, especially when working with logarithmic functions that result in decimal values. The exercise states the need to round the final result to three decimal places. Here’s how it is done:

• First, identify the first three decimal digits from your calculator result. For example, if you have a result like 0.895679, the digits are 895.
• Look at the fourth decimal place to decide rounding. In this case, it's '6'.
• If the digit is 5 or more, round the last decimal you keep up by one (895 becomes 896).
• If it's less than 5, keep the existing number as is.

This process ensures your answer is both precise and appropriate for mathematical and practical purposes. Rounding helps when you need a standardised number of significant digits, making it easier to report and compare results.