Problem 6 - Algebra 2

Question

Use a calculator to evaluate each expression to four decimal places. \(\ln 3.25\)

Step-by-Step Solution

Answer

The value of \( \ln 3.25 \) to four decimal places is 1.1787.

Step 1: Understanding the Natural Logarithm

The natural logarithm, denoted as \( ln\), is the logarithm to the base \( e \), where \( e \approx 2.71828 \). It is commonly used in mathematics and science when we study growth and decay processes.

Step 2: Input the Expression into a Calculator

To find \( ln 3.25 \), you need to use a scientific calculator capable of calculating natural logarithms. Make sure you enter the value 3.25 into the calculator and then press the \( ln \) function key.

Step 3: Understanding the Output

After entering \( 3.25 \) and the \( ln \) function, the calculator will provide an output. This value will be the natural logarithm of \( 3.25 \).

Step 4: Rounding to Four Decimal Places

The calculator should display a value that is approximately 1.178655. To round this to four decimal places, look at the fifth decimal place. Since it is a 5, you round up.

Step 5: State the Result

Hence, the natural logarithm of 3.25 to four decimal places is \( 1.1787 \).

Key Concepts

Mathematical Calculations

Mathematical calculations involving logarithms are a fundamental aspect of algebra that allows us to solve equations involving exponential growth or decay. The natural logarithm, denoted as \( \ln \), is particularly useful because it has the base \( e \), which approximately equals 2.71828. This base \( e \) is an irrational number and is crucial in many scientific and mathematical computations.
One of the key aspects of working with logarithms is understanding their properties, such as:

• \( \ln(ab) = \ln a + \ln b \)
• \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)
• \( \ln(a^b) = b \ln a \)

Using a natural logarithm can be efficient in solving problems related to exponential equations, as it helps convert them into linear equations, which are often easier to handle. This is why a lot of problems, including the one dealing with \( \ln 3.25 \), require a precise understanding of how to calculate and interpret natural logarithms.

Scientific Calculator

A scientific calculator is an essential tool for students working through various mathematical problems, such as finding the natural logarithm of a number. Unlike basic calculators, scientific calculators have a range of functions that allow you to calculate roots, powers, and logarithms.
Using a scientific calculator to find \( \ln 3.25 \) involves these simple steps:

• Switch on your scientific calculator and make sure it's set to the standard mode.
• Enter the number 3.25 using the number keypad.
• Locate the \( \ln \) button, normally placed with other log-related functions.
• Press the \( \ln \) button to calculate the natural logarithm.
• Read the displayed result which, in the case of \( \ln 3.25 \), would initially show a longer decimal like 1.178655.

Many scientific calculators also have features that allow you to immediately round off decimals, but understanding how to do this manually strengthens your grasp on mathematical accuracy.

Rounding Decimal Numbers

Rounding decimal numbers is a basic yet vital skill in mathematics that ensures accuracy and simplicity in reporting results. Once you've used a scientific calculator to find a natural logarithm or any other calculation, you often have to round the answer to a specific number of decimal places.
Here's how you can round off a number like 1.178655 to four decimal places:

• Identify the number at the fifth decimal place, which is 5 in this example.
• Because 5 is rounded upwards according to general rounding rules, you add 1 to the fourth decimal place.
• This turns a result of 1.178655 into 1.1787.

Rounding not only simplifies the number but also ensures it stays accurate enough for practical use. This is particularly important in scientific work, where reporting results with excessive decimal points might not only be unnecessary but could also lead to errors in further calculations if not handled correctly.