Problem 51 - College Algebra

Question

For the following exercises, evaluate the natural logarithmic expression without using a calculator. $$\ln (1)$$

Step-by-Step Solution

Answer

\( \ln(1) = 0 \).

Step 1: Understanding the Natural Logarithm

The natural logarithm, denoted as \( ext{ln}(x)\), is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. The expression inside the natural logarithm function in this problem is \(1\).

Step 2: Applying the Property of Logarithms

Recall the property of logarithms: the logarithm of \(1\) to any base is always \(0\) because any number raised to the power of \(0\) equals \(1\). Mathematically, this property is written as \( \ln(1) = 0 \).

Step 3: Write the Final Answer

From the property of logarithms, we conclude that the natural logarithm of \(1\) is \(0\). Therefore, the solution to the expression \( \ln(1) \) is \(0\).

Key Concepts

Logarithm Properties

Understanding the properties of logarithms helps simplify calculations and solve complex logarithmic equations. The primary property used in this case states that the logarithm of 1 to any base is always 0. This can be expressed mathematically as:

• For any base \( b \), \( \,\log_b (1) = 0 \)

This property holds true because no matter what the base is, raising it to the power of zero will always result in 1.
In addition to this, there are other useful logarithm properties, such as:

• The product property: \( \log_b (MN) = \log_b (M) + \log_b (N) \)
• The quotient property: \( \log_b \left( \frac{M}{N} \right) = \log_b (M) - \log_b (N) \)
• The power property: \( \log_b (M^p) = p \cdot \log_b (M) \)

These properties are fundamental concepts that you will frequently encounter and use as you become more comfortable with logarithmic expressions.

Evaluating Logarithms

Evaluating logarithms can often be simplified by using properties and breaking down the expression into more manageable parts. In the exercise, you are tasked with evaluating \( \ln(1) \) without a calculator.

Applying the logarithm property we discussed, you know that \( \ln(1) \) is 0 because any number to the power of 0 is 1.
Evaluating logarithms generally involves identifying relationships or utilizing properties that directly give you an answer or simplify your expression for easier computation.
For instance, if you know the base of the logarithm you’re dealing with, you can use known values and relationships:

• Logarithms of powers of the base are straightforward, for example \( \log_2 (8) = 3 \) because \( 2^3 = 8 \).
• Recognizing that \( \ln(e) = 1 \) can help simplify complex natural log evaluations.

Practice is key, and becoming familiar with these properties and operations will make evaluating logarithms a much simpler task.

Base e in Logarithms

Natural logarithms use \( e \) as the base, which is approximately 2.71828. This special number is integral to various areas of mathematics, particularly in calculus, because it simplifies the process of differentiation and integration in exponential functions.

The natural logarithm, denoted \( \ln(x) \), therefore refers to the logarithm with the base \( e \). You can think of \( \ln(x) \) as the power to which \( e \) must be raised to produce the number \( x \).
Just like how \( \log_{10}(x) \) uses 10 as its base, \( \ln(x) \) uniquely uses \( e \):

• If \( y = \ln(x) \), then \( e^y = x \).

In contexts where continuous growth or decay occur, such as population growth, radioactive decay, or interest calculation in finance, \( e \) and \( \ln(x) \) play significant roles.
Understanding this base is crucial for working with logarithms in these fields, and recognizing \( \ln(1) = 0 \) is a fundamental result easy to remember, thanks to its simplicity and importance in confirming base properties.