Problem 6 - Precalculus with Limits

Question

Fill in the blanks. The domain of the natural logarithmic function is the set of ______ ______ ______ .

Step-by-Step Solution

Answer

The domain of the natural logarithmic function is the set of positive real numbers.

Step 1: Identify the Function

The function given in the question is the natural logarithmic function, which is denoted as ln(x). The natural logarithm is the logarithm to the base 'e', where 'e' is a mathematical constant approximately equal to 2.71828. It is important to understand that the natural logarithm of x is the power to which e would have to be raised to equal x. For example: if \(x = e^y\), then \(y = ln x\).

Step 2: Know your Logarithms

In mathematics, a logarithm is undefined when the input value is less than or equal to zero. In other words, you can't take the logarithm of a negative number or zero. This is because the logarithm function is the inverse of the exponential function, and since an exponential function never equals zero, the logarithm is undefined at zero. Similarly, an exponential function only returns positive results, so the logarithm is undefined for negative numbers.

Step 3: Determine the Domain

Therefore, the domain of a logarithmic function includes all positive real numbers. The domain is the set of all valid inputs to the function. In this case, any positive real number can be input to the natural logarithm function. Thus, the words filling the blanks in the statement should be 'positive real numbers'.

Key Concepts

Domain of Logarithmic Function

The domain of a function refers to the set of all possible input values for which the function is defined. In the case of a logarithmic function, like the natural logarithm (denoted as \(\ln(x)\)), the domain is crucial to understand.
For the natural logarithmic function \(\ln(x)\), the domain is all positive real numbers. This means you can only input values greater than zero into the function.Using negative numbers or zero will result in an undefined value.

• Logarithms are inverses of exponential functions, which never yield negative or zero values.
• Therefore, the logarithm at these points is undefined.

This concept is essential because knowing the domain of the natural logarithm helps avoid errors in calculations and applications.

Logarithms

A logarithm is a mathematical concept that helps solve equations involving exponential functions. It provides the power to which a base number must be raised to obtain another number.
For example, if the base is 2 and the result you want is 8, you would solve for 3 because \( 2^3 = 8 \). So, the logarithm base 2 of 8 is 3.

• Logarithmic functions can have any positive number as a base, but natural logarithms specifically use the base \(e\), which is about 2.71828.
• The main property of logarithms is that they convert multiplication into addition, making computations simpler.

Understanding how logarithms work provides valuable insight into exponential growth and decay processes, which appear frequently in natural and scientific contexts.

Exponential Function

An exponential function is a mathematical function in the form \(f(x) = a^x\), where \(a\) is a constant called the base, and \(x\) is the exponent or power.
This function describes processes that change at a constant rate, such as compound interest, population growth, or radioactive decay.
Exponential functions have unique properties:

• They always yield positive results, no matter the value of \(x\).
• As \(x\) increases, the function values can grow extremely large.
• The inverse of an exponential function is the logarithmic function, which means that solving exponential equations often involves using logarithms.

Recognizing and applying the properties of exponential functions is key to solving complex mathematical and real-world problems.