Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ \frac{1}{e} $$

Step-by-Step Solution

Answer

The natural logarithm of \( \frac{1}{e} \) is approximately \(-1.00\).

Step 1: Recall the Property of Logarithms

Understand that the natural logarithm of a number is the power to which the base of the natural logarithm (which is Euler's number \( e \)) must be raised to obtain that number.

Step 2: Express the Number in Terms of e

Recognize that \( \frac{1}{e} \) can be rewritten as \( e^{-1} \). This makes it easier to calculate the natural logarithm because it directly involves the base \( e \).

Step 3: Use Logarithmic Identity

Use the identity \( \ln(a^b) = b \cdot \ln(a) \) to find \( \ln(e^{-1}) \). Here, \( a = e \) and \( b = -1 \). Therefore, \( \ln(e^{-1}) = -1 \cdot \ln(e) \).

Step 4: Compute the Logarithm of e

Since the natural logarithm of \( e \) is 1, it follows that \( \ln(e) = 1 \).

Step 5: Calculate the Natural Logarithm

Substitute the value from Step 4 into the equation in Step 3: \( \ln(e^{-1}) = -1 \times 1 = -1 \).

Step 6: Round to the Nearest Hundredth

Since \(-1\) is already a round number with two decimal places, the final result is \(-1.00\).

Key Concepts

Logarithmic Identities

Logarithmic identities are powerful tools in mathematics that help us manipulate logarithmic expressions to simplify calculations and solve equations. One of the most common identities is the power rule of logarithms, which states that the logarithm of a number raised to a power is equal to the exponent times the logarithm of the base:

• \( \ln(a^b) = b \cdot \ln(a) \)

This identity is particularly useful when dealing with natural logarithms, where the base is Euler's number, \( e \). It allows us to break down more complex expressions into simpler parts that are easier to calculate. For instance, using this identity, we can easily solve \( \ln(e^{-1}) \) by recognizing it as \( -1 \times \ln(e) \), simplifying the computation dramatically.Another commonly used identity is the product rule, which states:

• \( \ln(a \cdot b) = \ln(a) + \ln(b) \)

And the quotient rule, which is:

• \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)

These identities further enhance our ability to work with complex logarithmic expressions by breaking them into more manageable parts. By understanding and applying these identities, students can quickly resolve logarithmic problems and deepen their understanding of logarithmic functions.

Euler's Number

Euler's number, denoted by \( e \), is a fundamental constant in mathematics. It is approximately equal to 2.71828 and is the base of natural logarithms. The number \( e \) has important properties that are widely used in various fields such as calculus, complex analysis, and number theory.One of the defining characteristics of \( e \) is its natural growth properties. It arises naturally in the process of compounding interest continuously or when calculating limits that lead to exponential growth. For instance, when interest is compounded continuously, the formula \( A = P \cdot e^{rt} \) is used, where \( P \) is the principal amount, \( r \) is the rate, and \( t \) is time.In the realm of natural logarithms, \( e \) is significant because the natural logarithm is defined such that \( \ln(e) = 1 \). This property is crucial when working with exponential growth or decay problems, simplifying logarithmic computations dramatically. Euler's number also plays a key role in calculus, particularly when calculating derivatives and integrals of exponential functions.The importance of \( e \) cannot be overstated, as it serves as a cornerstone of higher-level mathematics. It provides a vital link between algebra, calculus, and natural growth processes in the real world.

Properties of Logarithms

Understanding the properties of logarithms is essential for solving logarithmic equations and simplifying expressions. These properties, which extend from the logarithmic identities, provide a framework for manipulating logarithmic expressions concisely.One vital property is the identity:

• \( \ln(e) = 1 \)

This property is foundational when dealing with natural logs because it means that the logarithm of the base itself (\( e \)) is straightforward: it equals 1. This simplifies expressions like \( \ln(e^{-1}) \) to \( -1 \), as multiply by \( 1 \) remains unchanged.Furthermore, the inverse property states:

• \( \ln(e^x) = x \)

This means that the natural logarithm and exponentiation by \( e \) are inverse operations; they cancel each other out. This property is used extensively in solving logarithmic and exponential equations.Additionally, the change of base formula allows conversion from one logarithmic base to another:

• \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \)

This formula can be especially helpful if you need to calculate logarithms on a different base but only have access to natural logs.These properties of logarithms make it easier to handle complicated logarithmic expressions, simplifying them into manageable parts, and ensure a smoother problem-solving process for students.