Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ 8.56 $$

Step-by-Step Solution

Answer

The natural logarithm of 8.56 is approximately 2.15.

Step 1: Identify the Function

To find the natural logarithm of a number, we will use the natural logarithm function, denoted as \( \ln(x) \). For our problem, we need to calculate \( \ln(8.56) \).

Step 2: Use a Calculator

Enter \( 8.56 \) into your scientific calculator and press the 'ln' button to compute the natural logarithm. Ensure your calculator is set to calculate natural logs, as opposed to base 10 logs.

Step 3: Obtain the Result

After computing \( \ln(8.56) \), the calculator will output a decimal value. For this exercise, the number you obtain is approximately \( 2.15 \) when rounded to the nearest hundredth.

Step 4: Rounding

Verify your result is rounded to two decimal places: For instance, if your calculator shows \( 2.14854 \), you will round it to \( 2.15 \) because 8 is the digit in the thousandth place which is 5 or more, rounding the hundredth place up.

Key Concepts

Using a Scientific Calculator

A scientific calculator is a must-have tool for many math problems, especially when dealing with logarithmic functions like natural logarithms. Unlike basic calculators, scientific calculators have advanced functions, such as logarithms, trigonometric functions, and exponents.

• When you want to calculate a natural logarithm, first make sure your calculator is turned on and set to calculate base 'e' logarithms.
• The 'ln' button is usually clearly marked. Enter the number whose natural logarithm you need and then press the 'ln' button.
• The calculator will display the result on the screen, making it an incredibly efficient tool for obtaining precise values.

If you struggle with finding functions on your calculator, consulting the manual specific to your model can be very helpful.

Understanding Rounding

Rounding is an essential mathematical skill, especially when dealing with decimals from calculator outputs. You round numbers to simplify them, often to make them easier to work with or understand.

• To round to the nearest hundredth as required in this exercise, look at the number in the third decimal place (the thousandth place).
• If this number is 5 or more, you increase the second decimal place by one. Otherwise, the second place stays the same.
• For example, if you calculate a number to be 2.14854, you would round this to 2.15, as the 8 in the third place prompts you to round up.

Remember, rounding does not change the value significantly but merely provides a practically usable form of the number.

Exploring Logarithmic Functions

Logarithmic functions are a key concept in mathematics, helping us understand exponential relationships.

• The natural logarithm, denoted as \( \ln(x) \), is a specific type of logarithm with a constant base 'e', where \( e \approx 2.718 \).
• It inversely relates to the exponential function \( e^x \), meaning that if \( y = e^x \), then \( \ln(y) = x \).
• This function compresses the scale of numbers, making very large values more manageable, which is why it frequently appears in scientific contexts.

Understanding the natural logarithm can be transformative, offering insights into diverse areas like decay processes, population growth, and even financial modeling.