Problem 51 - College Algebra

Question

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \ln x=2 $$

Step-by-Step Solution

Answer

The solution to the equation \( \ln x = 2 \) is \( x = e^2 = 7.39 \)

Step 1: Understand the given equation

The given equation is \( \ln x = 2 \). Here, \( \ln x \) stands for the natural logarithm, which is the logarithm to the base e, where e is approximately equal to 2.71828.

Step 2: Use properties of logarithms to solve for x

Using the principle that logarithm and exponential functions are inverses, we can rewrite the equation in the form \(e^{\ln x} = e^2\). This simplifies to \(x = e^2\).

Step 3: Evaluate and check the domain

Evaluating \(e^2\) gives approximately \(7.39\). We also check the domain of the original logarithmic expression which for \( \ln x \) is \( x > 0 \). Since \(7.39 > 0\), the solution is within the domain.

Step 4: Provide final answer

The solution to the given equation is \( x = e^2 = 7.39 \) (rounded to two decimal places)

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a special type of logarithm that has the mathematical constant \( e \) as its base. The constant \( e \) is approximately 2.71828 and is the foundation of natural exponential growth, appearing frequently in natural sciences and engineering. The natural logarithm is widely used in calculus and many other fields due to its unique properties that simplify derivatives and integrals involving exponential functions.

When you see \( \ln x \), it represents the power to which \( e \) must be raised to get \( x \). For example, when solving \( \ln x = 2 \), it means finding the value of \( x \) such that \( e^2 = x \).

Key properties of the natural logarithm include:

• \( \ln 1 = 0 \), because \( e^0 = 1 \).
• \( \ln e = 1 \), since \( e^1 = e \).
• The natural logarithm of a product: \( \ln(ab) = \ln a + \ln b \).
• The natural logarithm of a quotient: \( \ln \left(\frac{a}{b}\right) = \ln a - \ln b \).
• The natural logarithm of a power: \( \ln(a^b) = b \ln a \).

Understanding these properties can help simplify and solve logarithmic equations efficiently.

Domain of Logarithmic Functions

The domain of a logarithmic function like the natural logarithm is crucial as it determines the values for which the function is defined and meaningful. For the natural logarithm function \( \ln x \), the domain is defined as all positive real numbers, or mathematically speaking, \( x > 0 \).

This means that you cannot take the logarithm of zero or a negative number and expect a real number result. It's important to always check that the solutions obtained from logarithmic equations like \( \ln x = 2 \) fall within the established domain. In our example, since \( x = e^2 \), and \( e^2 \approx 7.39 \) which is greater than zero, the value is valid and within the domain.

Understanding and verifying the domain:

• Ensures the solution is valid and applicable.
• Prevents errors when solving logarithmic equations.
• Keeps calculations within the bounds of real numbers.

The adherence to domain constraints is essential when working with logarithmic and exponential functions.

Exponential Functions

Exponential functions, which are the inverse of logarithmic functions, have the form \( f(x) = a^x \), where \( a \) is a positive constant and \( x \) is any real number. When \( a = e \), we have the natural exponential function, written as \( e^x \), which exhibits continuous growth or decay.

The exponential function has the following main characteristics:

• It is always positive; \( e^x > 0 \) for any real number \( x \).
• It grows rapidly as \( x \) increases and approaches zero as \( x \) decreases.
• Its rate of growth, or derivative, is equivalent to its current value, making it unique: \( \frac{d}{dx}e^x = e^x \).

The connection between exponential functions and logarithms lies in their inverse relationship: If \( y = e^x \), then \( \ln y = x \). In solving \( \ln x = 2 \), we exploit this inverse relationship by expressing \( x \) as \( e^2 \).

This interplay between logarithms and exponential functions provides powerful tools in solving equations, modeling growth processes, and analyzing data across various fields.