Problem 37 - Precalculus Mathematics for Calculus

Question

Solve the logarithmic equation for \(x .\) $$\ln x=10$$

Step-by-Step Solution

Answer

The solution is \( x = e^{10} \approx 22026.4658 \).

Step 1: Convert from Logarithmic to Exponential Form

To solve the equation \( \ln x = 10 \), we first recognize that \( \ln x \) is the natural logarithm of \( x \), which is equivalent to \( \log_{e}x \). To convert this to exponential form, we rewrite the equation as \( x = e^{10} \).

Step 2: Calculate the value of \(e^{10}\)

Now that we have \( x = e^{10} \), we need to calculate this expression. Using a calculator, approximate \( e^{10} \). The value is approximately 22026.4658.

Step 3: Confirm the Solution

Verify the solution by substituting \( x = 22026.4658 \) back into the original logarithmic equation. Calculate \( \ln(22026.4658) \) to ensure that it approximately equals 10, confirming our solution is correct.

Key Concepts

Natural Logarithm

The natural logarithm is a mathematical concept that represents the logarithm to the base of the mathematical constant \( e \). Often represented as \( \ln \), it is extensively used in various fields of science and engineering due to its unique properties. The constant \( e \) is approximately equal to 2.71828, and it is an irrational number, meaning it cannot be expressed as a simple fraction.

When dealing with natural logarithms, remember that:

• \( \ln e = 1 \) because the natural logarithm of \( e \) is 1.
• \( \ln 1 = 0 \) since any base raised to the power 0 is 1.
• \( \ln(e^x) = x \) which indicates that the natural logarithm and exponential functions are inverses of each other.

This inverse relationship is fundamental and allows converting between logarithmic and exponential forms.
In our exercise, solving \( \ln x = 10 \) involves using this property to find the value of \( x \).

Exponential Form

The exponential form is a way to express logarithmic equations more straightforwardly. This transformation helps in solving logarithmic problems by converting them into exponents that are often easier to manage or compute. To convert a logarithm to an exponential form, you should use the base of the logarithm to raise it to the power of the opposite side of the equation.

For instance, our equation \( \ln x = 10 \) translates to exponential form as \( x = e^{10} \). This conversion highlights the relationship where if \( \ln x = y \), then \( x = e^y \).

• It simplifies the process of finding the solution by focusing on powers and base relationships.
• The exponential form is crucial in calculus, especially when dealing with growth models and differential equations.

By recognizing that \( \ln \) and \( e \) are inverses, you can efficiently solve logarithmic equations by transforming them into exponential ones, like we did in the original step-by-step solution.

Approximating e

The mathematical constant \( e \) is pivotal in both logarithmic and exponential equations, having significance beyond basic computations. "Approximating \( e \)" often involves using a calculator, as \( e \) is irrational, meaning it has an infinite amount of decimal places without repeating.

In scientific contexts, approximations of \( e \) are common to provide practical solutions. Importantly:

• \( e \) is approximately 2.71828, but for more precise calculations, you may need more decimal places.
• For \( e^{10} \), a calculator typically renders 22026.4658, as solved in the exercise.
• Accurate approximations are crucial when precision matters, such as in scientific calculations and predicting exponential growth.

Understanding how to approximate \( e \) is fundamental, especially when dealing with large powers like \( e^{10} \). In our solution, we used this approximation technique to find the value of \( x \).