Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the laws of logarithms to expand and simplify the expression. $$\ln \frac{e^{x}}{1+e^{x}}$$

Step-by-Step Solution

Answer

The simplified expression is \(\ln \frac{e^{x}}{1+e^{x}} = x - \ln(1+e^x)\).

Step 1: Apply the quotient rule of logarithms

Let's apply the first rule (quotient rule) to our expression: \(\ln \frac{e^{x}}{1+e^{x}} = \ln(e^x) - \ln(1+e^x)\)

Step 2: Apply the power rule of logarithms

Now, apply the second rule (power rule) to \( \ln(e^x) \): \(\ln(e^x) = x\cdot \ln(e)\) Since \( \ln(e) = 1\)(property of natural logarithm), the expression simplifies to: \(\ln(e^x) = x\)

Step 3: Substitute back into the expression

From steps 1 and 2, we have the expressions \( \ln \frac{e^{x}}{1+e^{x}} = \ln(e^x) - \ln(1+e^x)\) and \(\ln(e^x) = x\). Let's substitute the simplified expression for \( \ln(e^x)\) back into the first expression: \(\ln \frac{e^{x}}{1+e^{x}} = x - \ln(1+e^x)\) So, the expanded and simplified expression is: \(\ln \frac{e^{x}}{1+e^{x}} = x - \ln(1+e^x)\)

Key Concepts

Laws of Logarithms

Logarithms have rules that make them easier to work with, especially when simplifying complex expressions. These are known as the "laws of logarithms". Here are the primary ones you should know:

• Product Rule: The logarithm of a product is the sum of the logarithms. Mathematically, this is expressed as: \[\log_b(MN) = \log_b(M) + \log_b(N)\]
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms. This is written as: \[\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\]
• Power Rule: The logarithm of a power is the exponent times the logarithm of the base. It can be shown as: \[\log_b(M^n) = n\cdot \log_b(M)\]

In the given exercise, we apply both the quotient and power rules. The quotient rule lets us separate the fraction into a subtraction of two logs. Then, we use the power rule to simplify a log raised to a power.

Natural Logarithm

The natural logarithm is a specific type of logarithm with base \(e\), a mathematical constant approximately equal to 2.71828. It's written as \(\ln\) instead of \(\log_e\). The natural logarithm is particularly important in mathematics because it frequently occurs in growth models, calculus, and physics.

• Core Property: \(\ln(e)\) equates to 1, meaning any expression \(\ln(e^n)\) simplifies to \(n\) because of the power rule for logarithms: \[\ln(e^n) = n \cdot \ln(e) = n\]

In the exercise, recognizing that \(\ln(e^x)\) simplifies to \(x\) using this property is a crucial step. This simplification relies on the key characteristic of the natural logarithm—its relation to the exponential constant \(e\).

Logarithmic Identities

Logarithmic identities are equations that enable transformations and representations of logarithmic expressions in various forms. Besides the laws of logarithms, these identities can offer additional simplifications. Some basic logarithmic identities include:

• Identity Logarithm: \(\log_b(b) = 1\) for any positive base \(beq 1\). This identity is fundamental in the exercise, simplifying natural logarithm calculations.
• Logarithm of 1: \(\log_b(1) = 0\), as raising any positive number \(b\) to the power 0 results in 1.

Understanding these identities allows you to manipulate logarithmic expressions more efficiently. They help bridge gaps between different logarithmic concepts, enabling more intuitive solving approaches. The ability to employ these tools can significantly simplify otherwise complicated expressions.