Problem 49 - Precalculus

Question

Approximate (if possible) the common and natural logarithms using a calculator. Round to two decimal places. $$\ln 380$$

Step-by-Step Solution

Answer

\( \ln(380) \approx 5.94 \).

Step 1: Identify the Type of Logarithm

We need to calculate the natural logarithm of 380. Natural logarithms are written as \( \ln(x) \) where the base is the irrational number \( e \), approximately equal to 2.718.

Step 2: Use a Calculator

For a precise result, use a scientific calculator to find \( \ln(380) \). Enter 380 and press the \( \ln \) button. Ensure your calculator is set to output results in decimal form.

Step 3: Round the Result

The calculator will provide a long decimal number. Round this number to two decimal places to satisfy the requirement of the exercise.

Key Concepts

scientific calculator

A scientific calculator is an essential tool when dealing with complex mathematical operations such as logarithms. Unlike a basic calculator, it has advanced functions that can handle operations like trigonometric functions, factorials, and, importantly for this context, various types of logarithms.
To use a scientific calculator to find a natural logarithm, like \( \ln(380) \), you need to:

• Ensure your calculator is on and functioning in scientific mode.
• Enter the number for which you want to find the logarithm, in this case, 380.
• Press the \( \ln \) button, which tells the calculator to compute the natural logarithm.
• Wait for the calculator to display the result.

Sometimes, scientific calculators have a separate mode or button to switch between displaying answers in decimal or scientific notation. Being familiar with these functions can save you time and ensure accuracy when solving mathematical problems.

rounding decimals

Rounding decimals is a necessary skill when you want to present a more readable number, especially in exercises requiring precision to a specified number of decimal places. In our case, you need to round the result of \( \ln(380) \) to two decimal places.
Here’s a simple process to effectively round a decimal number:

• Look at the third decimal number (if you are rounding to two decimal places) to decide whether to round up or down.
• If this number is 5 or more, increase the second decimal place by one.
• If this number is less than 5, keep the second decimal place as it is.

For instance, if your calculator gives you 5.940, rounding to two decimal places would result in 5.94 because the third decimal is 0, which means the number stays the same.

base e

The natural logarithm is a logarithmic function where the base is the mathematical constant \( e \). This number is approximately equal to 2.718, and it is an irrational and transcendental number. It emerges naturally in many mathematical contexts.
Here's why \( e \) is significant in natural logarithms:

• It represents the unique rate of growth shared by all continuously growing phenomena.
• It is the base of the natural logarithm, noted as \( \ln \), which is commonly used in calculus, complex numbers, and finance.
• Using base \( e \) allows for a more natural way of describing exponential change especially pertinent to real-world applications.

Understanding the concept of base \( e \) helps make sense of why natural logarithms are used frequently in both theoretical and applied mathematics.