Problem 11 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To find \(\ln e^{2},\) we ask, "To what power must we raise to get \(e^{2} ? "\) since the answer is the 2 nd power, \(\ln e^{2}=\) ___.

Step-by-Step Solution

Answer

The solution to \( \ln e^{2} = 2 \).

Step 1: Understanding Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the inverse of the exponential function. Specifically, \( \ln(e^x) = x \). This means that the logarithm of an exponential power simplifies to the exponent itself.

Step 2: Applying the Logarithm Property

Using the logarithm property from Step 1, apply it to the expression \( \ln(e^2) \). This simplifies directly because \( \ln(e^x) \) simplifies to \( x \).

Step 3: Simplifying the Expression

Given \( \ln(e^2) \), we use the property learned that transforms it to the exponent. Therefore, \( \ln(e^2) = 2 \).

Key Concepts

Inverse Functions

An inverse function is a fundamental concept in mathematics, where a function essentially "undoes" the action of another function. In simpler terms, if you have a function that takes you from point A to point B, the inverse function will bring you back from point B to point A. For example, if you add 2 to a number using a function, the inverse function would subtract 2 to get you back to the original number.

In the world of functions, the inverse has a special notation. If your original function is identified as \( f(x) \), its inverse is written as \( f^{-1}(x) \). When you apply the function and then its inverse, you'll end up where you started:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

In the context of logarithms, the natural logarithm \( \ln(x) \) is the inverse of the exponential function \( e^x \). This means that if you exponentiate a number and then take the natural logarithm of the result, you end up with the original number.

The relationship between inverse functions is an essential tool for solving equations and understanding function behavior.

Exponential Functions

Exponential functions are a type of mathematical function where a constant number, called the base, is raised to a variable exponent. These functions are vital in many fields, including biology, finance, and physics, due to their ability to model growth patterns and decay processes. A typical exponential function looks like this: \( f(x) = a^x \), where \( a \) is the base.

One of the most significant exponential functions is \( f(x) = e^x \), where \( e \) is a mathematical constant approximately equal to 2.71828. This particular function is unique because its derivative is the same as the function itself. This property makes it extremely valuable in calculus, simplifying many complex calculations.

Key characteristics of exponential functions include:

• Rapid growth or decay: Depending on whether the exponent is positive or negative.
• A consistent growth rate: The percentage rate of increase remains constant over time.
• Positive outputs: Exponential functions always yield positive results, as long as the base is positive.

These characteristics make exponential functions powerful tools for predicting future values and understanding how quantities evolve over time.

Logarithmic Properties

Logarithms are the mathematical inverse of exponential functions. They allow us to solve equations where the unknown variable is an exponent. The logarithm gives us the power to which a base number must be raised to get another number. For example, if \( b^y = x \), then \( \log_b(x) = y \).

The properties of logarithms are essential for simplifying and solving expressions involving powers. Some fundamental logarithmic properties include:

• Product Property: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Property: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Property: \( \log_b(M^k) = k \cdot \log_b(M) \)
• Change of Base Formula: \( \log_b(M) = \frac{\log_c(M)}{\log_c(b)} \)

Each of these properties allows us to manipulate and solve logarithmic equations more effectively. In the context of natural logarithms, we deal specifically with the base \( e \), with the notation \( \ln(x) \).

In solving problems, recognizing logarithmic properties is key to simplifying complex expressions and finding solutions efficiently.