Problem 60 - 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. $$\ln (-10)$$

Step-by-Step Solution

Answer

The expression \(\ln(-10)\) is undefined for real numbers.

Step 1: Understanding Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is defined only for positive real numbers \(x > 0\). This means that it cannot be evaluated for negative numbers or zero.

Step 2: Analyzing the Given Expression

The exercise asks to evaluate \(\ln(-10)\). Since "-10" is a negative number, \(\ln(-10)\) is undefined in the context of real numbers.

Step 3: Conclusion Based on Definitions

Given the restrictions on the domain of the natural logarithm function, \(\ln(-10)\) does not have a real number value. Thus, the expression cannot be evaluated using a typical calculator that operates in the real number system.

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions used to find the power to which a base, usually 10 or the natural base, must be raised to produce a given number. The natural logarithm, denoted as \( \ln(x) \), is a type of logarithm that uses the base \( e \) (approximately 2.718). This function is crucial because it provides a way to model growth and decay processes, which are common in various scientific fields.
Logarithmic functions are the inverse of exponential functions. If you know that \( e^y = x \), then \( \ln(x) = y \). This inversion is the key characteristic that defines and distinguishes logarithmic functions.
The important aspect of any logarithmic function is understanding its base and input values. Bases determine the scale of the growth rates in exponential terms, and they must always be positive. That constraint often introduces limitations to the domain of logs, which is an essential concept to grasp.

Domain and Range

The domain and range are vital in understanding mathematical functions, including logarithmic functions like the natural logarithm. In simple terms, the domain of a function is the complete set of possible input values, while the range is the set of possible output values.
When it comes to the natural logarithm, \( \ln(x) \), the domain is restricted to positive real numbers \( x > 0 \). Consequently, since the natural log function is only defined for positive numbers, inputting a negative number or zero isn't feasible. This is because there's no real power to which \( e \) can be raised to yield a negative result or zero.

• Domain of \( \ln(x) \): Positive real numbers \( (0, \infty) \)
• Range of \( \ln(x) \): All real numbers \( (-\infty, \infty) \)

Understanding these concepts helps to clarify why \( \ln(-10) \) is undefined in the real number system.

Calculator Usage

Using a calculator is a valuable skill when it comes to evaluating expressions quickly and accurately. For logarithmic functions, most scientific calculators have dedicated keys like \( \ln \) and \( \log \) for natural logarithm and base-10 logarithm calculations.
When you attempt to input a negative number into the \( \ln \) function on a typical calculator, it will return an error or undefined message. This error highlights that the function is outside its domain of positive real numbers.
Here are some tips for using calculators in logarithmic functions:

• Ensure the input falls within the domain. For \( \ln \), use only positive numbers.
• Verify calculator settings are set to the correct mode (often degree or radian, but this is more for trigonometry).
• Understand potential errors by referring back to mathematical definitions and properties.

These pointers will help you avoid confusion and misuse, especially when dealing with natural logarithms.