Problem 183 - Precalculus

Question

Is there a number \(x\) such that \(\ln x=2 ?\) If so, what is that number? Verify the result.

Step-by-Step Solution

Answer

Yes, the number is \(e^2 \approx 7.389\).

Step 1: Understand the Equation

The equation gives us \ \ln(x) = 2 . \ Our task is to find a number \(x\) such that when we apply the natural logarithm to \(x\), the result is 2.

Step 2: Use the Property of Natural Logarithms

The natural logarithm, denoted as \(\ln\), is the inverse operation of the exponential function with base \(e\). Therefore, the equation \(\ln(x) = 2\) can be rewritten using exponents: \ x = e^2.

Step 3: Calculate the Exact Value

Now that we have expressed \(x\) in terms of \(e\), we calculate its value: \ x = e^2. \ The value of \(e\) (Euler's number) is approximately 2.718. Thus, \ e^2 \approx 2.718^2 \approx 7.389.

Step 4: Verify the Result

To verify, we substitute \(x = e^2\) back into the original equation and check: \ \ln(e^2) = 2 \ Using the property of logarithms, \(\ln(a^b) = b\cdot\ln(a)\), we get: \ \ln(e^2) = 2\cdot\ln(e) = 2\cdot1 = 2. \ This confirms that our solution is correct.

Key Concepts

Exponential Function

An exponential function is a mathematical function of the form \( f(x) = a^x \), where \( a \) is a positive constant known as the base, and \( x \) is the exponent. In many real-world scenarios, such functions model growth or decay processes, where the change rate is proportional to the current value. In our exercise, the base of the exponential function is Euler's number \( e \), leading to the function \( e^x \). This particular form is highly significant in mathematics.
The function \( e^x \) is special as it is its own derivative and also its own integral, up to constant factors. This makes it widely applicable in calculus, especially in solving differential equations. Understanding exponential functions is crucial because they lay the foundation for comprehending logarithmic functions, particularly the natural logarithm, which is the focus of our exercise.

• The inverse of an exponential function is the logarithmic function.
• Exponential growth can describe things like population size or investment over time.

Here, determining \( x \) such that \( \ln(x) = 2 \) involved switching from a logarithmic perspective to an exponential one, utilizing the concept \( x = e^2 \).

Logarithmic Properties

Logarithms help us solve exponential equations by allowing us to "undo" exponentiation. The natural logarithm \( \ln \) is a logarithm with base \( e \), and it tells us the power to which \( e \) must be raised to obtain a certain number. For example, if \( \ln(x) = 2 \), it means \( e^2 = x \).
Logarithmic properties are essential for simplifying complex multiplication and division into more manageable addition and subtraction, which arise frequently in electronics, sound scales, and financial calculations. Here are some useful logarithmic properties:

• \( \ln(a \cdot b) = \ln a + \ln b \)
• \( \ln\frac{a}{b} = \ln a - \ln b \)
• \( \ln(a^b) = b \cdot \ln a \)

By applying these properties, we verified that substituting \( x = e^2 \) back into \( \ln(x) \), we indeed obtained the value 2, confirming the solution's correctness in the original exercise.

Euler's Number

Euler's number, denoted as \( e \), is a fundamental constant approximately equal to 2.71828. It plays a pivotal role in mathematics, especially in calculus, complex numbers, and exponential growth processes. Introduced by Swiss mathematician Leonhard Euler, \( e \) has properties that make it unique among numbers.
In exponential functions, \( e \) serves as the base for the natural exponential function \( e^x \). This function's rate of growth is proportional to its current value, a characteristic vital in continuous growth processes like radioactive decay and compound interest calculations.\( e \) is defined as the limit of \( (1 + \frac{1}{n})^n \) as \( n \) approaches infinity, and it appears in many areas of mathematics:

• It connects logarithms and trigonometric functions through Euler's formula: \( e^{ix} = \cos(x) + i\sin(x) \).
• It represents the base of the natural log function.
• Is essential in differential equations as solutions often involve \( e \).

In the context of the exercise, identifying \( x \) as \( e^2 \) directly relies on understanding \( e \) and its role in exponential and logarithmic functions.