Problem 55 - Elementary and Intermediate Algebra

Question

Use a calculator to evaluate each expression, if possible. Express all answers to four decimal places. See Using Your Calculator: Evaluating Base-e (Natural) Logarithms. $$\ln 0.00465$$

Step-by-Step Solution

Answer

\( \ln(0.00465) \approx -5.3682 \)

Step 1: Recognize the Expression

The expression to be evaluated is the natural logarithm of the number 0.00465, which is written as \( \ln(0.00465) \). Natural logarithms use base \( e \), where \( e \) is approximately 2.71828.

Step 2: Input the Expression into the Calculator

To find \( \ln(0.00465) \), turn on your calculator and find the \( \ln \) or naturally labeled 'LN' button. Enter 0.00465, then press the \( \ln \) button to compute the natural logarithm.

Step 3: Record the Result

The calculator should display the result after you press the \( \ln \) button. Make sure the calculator is in the correct calculation mode to display decimal results.

Step 4: Round to Four Decimal Places

The calculator will give a result which may have more than four decimal places. Round the result to four decimal places for the final answer.

Key Concepts

Understanding the Base e

In mathematics, many exponential and logarithmic functions operate on a special number known as the base \( e \). The number \( e \), approximately 2.71828, is an irrational number. This means it cannot be expressed as a simple fraction. The base \( e \) has numerous applications in the real world, especially in areas involving growth and decay, such as biology, finance, and physics. What's fascinating is that \( e \) frequently appears in calculations involving continuous growth processes. Natural logarithms make use of this base \( e \), simplifying equations and making them more applicable in real-life scenarios. Remember, whenever you see a natural logarithm, you’re working with this special number \( e \).

Moreover, because \( e \) is a fundamental constant, understanding its behavior can help make complex mathematical models easier to understand and manage.

Exploring Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. When we talk about natural logarithms, we're referring to logarithms with the base \( e \). A natural logarithm asks the question: to what exponent must \( e \) be raised to result in a particular number? For example, \( \ln(x) \) answers the question: "What power should \( e \) be raised to, to produce \( x \)?"

These functions are pivotal in solving equations where the unknown is in an exponential position. By using the properties of logarithms, complex multiplicative processes become more straightforward to handle. This kind of function is prevalent in solving real-life problems involving scaling and natural growth, from calculating interest rates in finance to population growth modeling.

Calculator Usage for Natural Logarithms

Using a calculator can significantly simplify computing natural logarithms, especially when dealing with non-whole numbers. Most scientific calculators have a dedicated button marked as ‘LN’ for this purpose. To compute \( \ln(x) \), simply:

• Turn on your calculator.
• Enter the value of \( x \), like 0.00465.
• Press the ‘LN’ button to find the natural logarithm.

Calculators are usually set to compute in decimals, ensuring that you gain accurate results that would be tedious to calculate by hand. However, ensure that your calculator is in the right mode for logarithmic functions. This method speeds up the computational process and reduces the risk of human error.

Rounding to Four Decimal Places

Rounding numbers is a common practice to ensure that results are practical and consistent, especially in complex calculations. Once your calculator outputs the value of a natural logarithm, it might display many decimals. For clarity and precision, the result is typically rounded to four decimal places.

To round to four decimal places, look at the fifth digit:

• If it's 5 or greater, round up the fourth digit by one.
• If it's less than 5, keep the fourth digit as it is.

For the expression \( \ln(0.00465) \), if the calculator shows \(-5.36949\), it will be rounded to \(-5.3695\). Such rounding makes the result easier to interpret and prevents information overload, while still maintaining an acceptable level of accuracy.