Problem 14 - Calculus

Question

Solve for \(x\) using logs. $$2 e^{3 x}=4 e^{5 x}$$

Step-by-Step Solution

Answer

\(x = -\frac{1}{2} \ln(2)\)

Step 1: Simplify the Expression

Start by dividing both sides of the equation by \(e^{3x}\). This gives us \(2 = 4 e^{2x}\).

Step 2: Isolate the Exponential Term

Divide both sides by 4 to get the exponential term by itself: \(\frac{1}{2} = e^{2x}\).

Step 3: Apply the Natural Logarithm

Take the natural logarithm (\(\ln\)) of both sides. We have \(\ln\left(\frac{1}{2}\right) = \ln\left(e^{2x}\right)\).

Step 4: Simplify Using Logarithmic Identity

Use the identity \(\ln(e^y) = y\) to simplify the right-hand side: \(\ln\left(\frac{1}{2}\right) = 2x\).

Step 5: Solve for x

Divide both sides by 2 to isolate \(x\): \(x = \frac{1}{2} \ln\left(\frac{1}{2}\right)\).

Step 6: Express x in Terms of Known Logarithms

Recognize that \(\ln\left(\frac{1}{2}\right) = -\ln(2)\), so \(x = -\frac{1}{2} \ln(2)\).

Key Concepts

Understanding Natural Logarithms

The natural logarithm, denoted as \(\ln(x)\), is a special type of logarithm that has a base of the mathematical constant \(e\), which is approximately 2.71828. This kind of logarithm is very common in mathematics because of its unique properties that make solving equations involving exponential functions more straightforward. Here's why it's important:

• Natural logarithms are used because they make the process of dealing with complex exponential expressions easier.
• The natural logarithm of \(e^x\) is simply \(x\) due to the identity \(\ln(e^x) = x\). This powerful property simplifies solving exponential equations.

Whenever you see an exponential function, applying the natural logarithm can help you transform the equation into a simpler form, making it easier to solve. In our exercise, applying \(\ln\) allowed us to tackle the equation \( \ln\left(e^{2x}\right) \), directly simplifying it to \(2x\). This is why understanding and utilizing natural logarithms are crucial in solving mathematical problems involving exponential growth or decay.

Exploring the Exponential Function

The exponential function \(e^x\) is a mathematical function expressed in terms of the base \(e\), which is a constant approximately equal to 2.71828. It's a fundamental concept in mathematics, especially for modeling situations where changes occur at a consistent rate.

• An exponential function involves a constant raised to a power, where the power itself is a variable.
• These functions are commonly used in real-life applications like compound interest, population growth, and radioactive decay.

In our exercise, the function \(e^{3x}\) and \(e^{5x}\) appear on both sides of the equation. By dividing both sides by \(e^{3x}\), we transform the equation, focusing on the remaining exponential relation. This simplification is a crucial step in solving the problem as it reduces the equation to a form we can easily handle with logarithms, specifically the natural logarithm. Understanding how to manipulate exponential functions is key to simplifying and solving these equations.

Techniques for Solving Equations Using Logarithms

Solving equations using logarithms involves specific techniques that rely on understanding both the properties of logarithms and the behavior of exponential functions. Here's a step-by-step guide to how these techniques were applied to our exercise:

• Simplify the Equation: Always start by simplifying the expression as much as possible. In our case, we divided both sides by \(e^{3x}\) to make the equation easier to handle.
• Isolate the Exponential Part: Focus next on isolating the exponential term by using algebraic operations. For example, we divided by 4 to isolate \(e^{2x}\).
• Apply the Natural Logarithm: Use \(\ln\) to both sides of the equation, which helps in getting rid of the exponential part and uncovering the variable. This transformed \( e^{2x} \) into a linear form \(2x\).
• Solve for the Variable: Now you're left with a basic algebra equation, \(2x = \ln\left(\frac{1}{2}\right)\). Solving for \(x\) becomes simple using basic division.

These techniques illustrate how logarithms can turn complex-looking equations into manageable ones. By consistently applying these steps, you can solve a wide range of problems involving exponential expressions.