Problem 41 - Intermediate Algebra

Question

Evaluate each expression without using a calculator. $$ \ln e^{5} $$

Step-by-Step Solution

Answer

\( \ln e^5 = 5 \).

Step 1: Understand the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm with base \( e \), where \( e \) is approximately equal to 2.71828. The expression \( \ln e^5 \) asks for the exponent to which \( e \) must be raised to produce \( e^5 \).

Step 2: Use the Property of Logarithms

A key property of logarithms is that \( \ln e^x = x \). This means that when you take the natural logarithm of \( e \) raised to any power, you simply obtain the exponent. Applying this property to our expression, \( \ln e^5 = 5 \).

Step 3: Final Evaluation

According to the logarithmic property used, the expression evaluates to the exponent itself, so \( \ln e^5 = 5 \).

Key Concepts

Logarithmic Properties

Logarithms help to simplify complex mathematical expressions by turning multiplication into addition and exponentiation into multiplication. They are particularly useful when handling operations involving large numbers or powers. A fundamental property of logarithms is the relationship between exponentiation and logarithms, especially with natural logarithms. This property states that if you have a number in the form of \( \ln(e^x) \), it simplifies directly to \( x \). This is because logarithms are the inverse operations of exponentiation. Hence, when you take the logarithm of something already exponentiated by that base, you return to the exponent itself.

• The logarithm of a product is the sum of the logarithms.
• The logarithm of a quotient is the difference of the logarithms.
• The logarithm of a power means you can bring the exponent in front of the logarithm.

When working with expressions like \( \ln(e^5) \), leveraging these properties helps simplify the calculations, saving time and avoiding errors.

Exponentiation

Exponentiation is raising a number to a certain power. The expression \( e^5 \) is an example of exponentiation. Here, \( e \) is the base and \( 5 \) is the exponent. The result of this operation is the number \( e \) multiplied by itself 5 times. Exponentiation is a fundamental mathematical operation that extends the idea of repeated multiplication. Understanding this concept is crucial when dealing with logarithmic expressions, as they often involve reversing the process of exponentiation.

• The base tells you what number to multiply.
• The exponent tells you how many times to use the base in a multiplication.

By comprehending exponentiation, you are well-equipped to apply it alongside logarithms to break down and solve complex expressions effortlessly.

Mathematical Expressions

Mathematical expressions involve combining numbers and operations to express a value or relationship. In our context, \( \ln(e^5) \) is a concise way to represent the relationship between logarithms and exponentiation. It includes key mathematical elements like the constant \( e \), which is crucial in natural logarithms. Simplifying such expressions relies heavily on understanding both the logarithmic properties and exponentiation rules.

• An expression like \( \ln(e^5) \) involves recognizing how the natural log and exponent function cancel each other out.
• It showcases the elegance of mathematical operations working together.

Ultimately, mastering how to evaluate and simplify mathematical expressions requires practice with these foundational concepts, enabling you to tackle a wide range of problems with confidence.