Problem 6 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ e^{2}=7.3891 \ldots $$

Step-by-Step Solution

Answer

The logarithmic form of the given exponential equation is \(ln(7.3891) = 2\).

Step 1: Analyze the exponential equation

The provided equation is an exponential equation in the form of \(a^{b} = c\), where \(a\) is the base (in this case, \(e\)), \(b\) is the exponent (in this case, 2), and \(c\) is the result (approximately 7.3891).

Step 2: Apply the properties of logarithms

According to the properties of logarithms, an exponential equation \(a^{b} = c\) can be translated into a logarithmic equation as \(\log_{a}c = b\). This means the base of the logarithm will be \(a\), the input of the logarithms will be \(c\), and the output will be \(b\).

Step 3: Convert the exponential equation into a logarithmic equation

Following the principles outlined in Step 2, the new logarithmic equation will become: \(\log_{e} (7.3891) = 2\). Is common to write the logarithmic equation with base \(e\) as a natural logarithm \(ln\), the equation will be: \(ln(7.3891) = 2\).

Key Concepts

Exponential Functions

Exponential functions are an essential concept in mathematics. They consist of functions in which a constant base is raised to a variable exponent. In the equation given above, the base is the mathematical constant known as Euler's number, denoted as \(e\). Here are the main characteristics of exponential functions:

• The base is a positive real number, typically greater than 0.
• The base is raised to an exponent, which can be any real number.
• Exponential growth occurs when the base is greater than 1, while exponential decay happens if the base is between 0 and 1.
• Exponential functions exhibit rapid growth due to the properties of exponents.

In our exercise, the given exponential equation is \(e^2 = 7.3891\). This tells us that when Euler's number \(e\) is raised to the power of 2, the resulting value is approximately 7.3891. Understanding exponential functions is crucial for solving equations that involve these rapidly growing terms.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They help in solving equations where the unknown is in the exponent. Any exponential equation can be rewritten as a logarithmic equation.

• A logarithm asks the question: "To what power must we raise the base to obtain a given number?"
• The base of the logarithm is the same as the base of the corresponding exponential function.
• The result of the logarithm is the exponent from the exponential equation.

In the problem provided, the exponential equation \(e^2 = 7.3891\) was rewritten as a logarithmic equation: \(\log_{e}(7.3891) = 2\). This format tells us that \(e\) raised to the power of 2 equals 7.3891. Logarithmic functions assist in determining the exponent that achieves a certain value with a given base.

Natural Logarithm

The natural logarithm is a special type of logarithm where the base is the natural number \(e\), approximately equal to 2.71828. It's denoted as \(\ln(x)\). The natural logarithm is widely used in mathematics, physics, and engineering due to its unique properties:

• It is the inverse operation of raising \(e\) to a power.
• Natural logs simplify multiplicative processes because of their additive properties. If \(a = e^x\), then \(\ln(a) = x\).
• They have important applications in calculating continuous growth or decay, often found in financial math, population dynamics, and chemical reactions.

In this exercise, the exponential equation \(e^2 = 7.3891\) is expressed conveniently as a natural logarithmic equation: \(\ln(7.3891) = 2\). This not only aligns with how logs and exponentials are inverses but also showcases the power and symmetry of natural logarithms in solving real-world problems.