Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of natural logarithms to rewrite the expression. $$\ln e^{2}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is 2.

Step 1: Apply logarithmic property

We know that if we have \(\ln a^b\), it can be written as \(b \ln a\). Using this property, we can simplify the given expression as \(2 \ln e\).

Step 2: Use logarithm identity

The natural logarithm of \(e\) is 1. So, \(2 \ln e\) becomes \(2 \times 1 = 2\). After this step, the entire problem is solved.

Key Concepts

Logarithmic Properties

Logarithms are an integral part of mathematics, particularly when dealing with exponential equations. Understanding logarithmic properties is fundamental in simplifying complex expressions into more manageable forms.

The most crucial logarithmic properties to remember include:

• The product rule: \( \ln(ab) = \ln(a) + \ln(b) \), which allows us to separate logarithms of multiplications into sums.
• The quotient rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \), which enables us to write the logarithm of a division as the difference of logarithms.
• The power rule: \( \ln(a^b) = b\ln(a) \), which is essential for handling powers within logarithms by bringing the exponent out as a coefficient.

Applying these properties can break down complex expressions and provide a pathway to solving logarithmic equations. For example, in the textbook exercise, by identifying and applying the power rule \( \ln(e^2) = 2\ln(e) \) the expression becomes simpler to resolve.

Natural Logarithm Identities

The natural logarithm, represented as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. A deep understanding of the natural logarithm identities is key to mastering the art of simplifying logarithmic expressions.

Some of the most pivotal identities include:

• \(\ln(1) = 0\): The log of 1 to any base is always 0 because any number raised to the power of 0 equals 1.
• \(\ln(e) = 1\): The log of \(e\) to the base \(e\) is 1, since the definition of the logarithm is to find the exponent that the base must be raised to produce a given number.
• \(\ln(e^x) = x\): The natural log of \(e \text{ raised to any exponent } x\) simply returns \(x\).

In the exercise provided, the identity \(\ln(e) = 1\) is used in the second step to simplify \(2\ln(e)\) to \(2\times1\) hence obtaining the answer 2.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using logarithmic properties and identities to condense or rewrite expressions for ease of understanding and computation.

To effectively simplify a logarithmic expression:

• Identify the log properties that can be applied.
• Rewrite the equation using these properties.
• Simplify further by substituting known logarithm values and performing basic arithmetic operations.

In practice, the exercise solution takes the given \(\ln(e^2)\) and simplifies it using the power rule, resulting in \(2\ln(e)\). Recognizing that \(\ln(e)\) is equal to 1, the expression simplifies to 2. This methodical approach can be applied to a multitude of logarithmic expressions for efficient problem-solving.