Solving The Exponential Equation E^(3x+2) = 1/e^4 A Step-by-Step Guide

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In this article, we will delve into the step-by-step process of solving the exponential equation e^(3x+2) = 1/e^4. Exponential equations, where the variable appears in the exponent, are a crucial topic in mathematics, frequently encountered in algebra, calculus, and various scientific applications. Our goal is to find the exact solutions for x that satisfy this equation. We will start by manipulating the equation using properties of exponents, then apply logarithms to isolate the variable, and finally, arrive at the solution set. Understanding how to solve such equations is fundamental for anyone studying mathematics or related fields. Let's embark on this mathematical journey and solve this equation methodically.

Before we dive into the specific steps for solving e^(3x+2) = 1/e^4, it's essential to grasp the core concepts of exponential equations. An exponential equation is an equation in which a variable occurs in the exponent. The base, in this case e (Euler's number, approximately 2.71828), is a constant. The key to solving these equations lies in understanding and applying the properties of exponents and logarithms. Logarithms are the inverse operation to exponentiation, allowing us to "undo" the exponent and solve for the variable.

In our equation, e^(3x+2) = 1/e^4, the left side has the exponential term e^(3x+2), while the right side can be rewritten using the properties of exponents. Specifically, we know that 1/e^4 can be expressed as e^(-4). This conversion is a crucial first step because it allows us to equate the exponents directly once we have the same base on both sides of the equation. The properties of exponents, such as a^(-n) = 1/a^n and a^m = a^n implies m = n, are vital tools in solving exponential equations. Recognizing these properties and applying them correctly is the foundation for finding the solution.

To solve the equation e^(3x+2) = 1/e^4, we'll follow a series of methodical steps, applying the properties of exponents and logarithms to isolate x. Each step is designed to bring us closer to the solution, ensuring clarity and accuracy in our process.

Step 1: Rewrite the Equation

The first step involves rewriting the equation to have the same base on both sides. As mentioned earlier, we can express 1/e^4 as e^(-4). This transformation allows us to rewrite the equation as:

e^(3x+2) = e^(-4)

By doing this, we've set the stage for equating the exponents, which is a crucial technique in solving exponential equations.

Step 2: Equate the Exponents

Now that we have the same base on both sides of the equation, we can equate the exponents. This is based on the property that if a^m = a^n, then m = n. Applying this property to our equation, we get:

3x + 2 = -4

This simplifies the problem significantly, transforming the exponential equation into a simple linear equation.

Step 3: Solve the Linear Equation

We now have a linear equation, 3x + 2 = -4, which can be solved using basic algebraic techniques. First, we subtract 2 from both sides of the equation:

3x + 2 - 2 = -4 - 2

3x = -6

Next, we divide both sides by 3 to isolate x:

(3x)/3 = (-6)/3

x = -2

Thus, we have found the value of x that satisfies the equation.

Step 4: Verify the Solution

It's always a good practice to verify the solution by plugging it back into the original equation. Substituting x = -2 into the original equation e^(3x+2) = 1/e^4, we get:

e^(3(-2)+2) = 1/e^4

e^(-6+2) = 1/e^4

e^(-4) = 1/e^4

Since e^(-4) is indeed equal to 1/e^4, our solution x = -2 is correct.

The solution set for the equation e^(3x+2) = 1/e^4 is the set of all values of x that satisfy the equation. In this case, we found only one solution, x = -2. Therefore, the solution set can be written as:

{-2}

This set contains the single, exact solution to the given exponential equation. The solution set is a concise way to represent all possible solutions and ensures that we've accurately captured the answer.

While we solved e^(3x+2) = 1/e^4 by equating exponents, it's worth noting that other methods can be used, particularly when dealing with more complex exponential equations. One such method involves using logarithms directly.

Using Logarithms

If we start with the original equation e^(3x+2) = 1/e^4, we can take the natural logarithm (ln) of both sides. The natural logarithm is the logarithm to the base e, and it is particularly useful when dealing with exponential functions involving e because of the property ln(e^x) = x.

Taking the natural logarithm of both sides, we get:

ln(e^(3x+2)) = ln(1/e^4)

Using the property of logarithms that ln(a^b) = b * ln(a), we simplify the left side:

(3x+2) * ln(e) = ln(1/e^4)

Since ln(e) = 1, we have:

3x + 2 = ln(1/e^4)

We can rewrite 1/e^4 as e^(-4), so:

3x + 2 = ln(e^(-4))

Using the logarithm property again, we get:

3x + 2 = -4 * ln(e)

Since ln(e) = 1:

3x + 2 = -4

From this point, we proceed exactly as before, subtracting 2 from both sides and then dividing by 3 to get:

3x = -6

x = -2

This method confirms our earlier solution, demonstrating the versatility of using logarithms in solving exponential equations. This approach is especially valuable when dealing with more complex exponents or when it's not straightforward to equate the exponents directly.

Exponential equations are not just theoretical constructs; they have significant real-world applications across various fields. Understanding how to solve them is crucial for analyzing and predicting phenomena in science, engineering, finance, and more.

Population Growth

One of the most common applications of exponential equations is in modeling population growth. The growth of a population (whether it's bacteria in a culture, a species in an ecosystem, or the human population) can often be described by an exponential function. The equation typically takes the form:

P(t) = P_0 * e^(kt)

Where:

  • P(t) is the population at time t,
  • P_0 is the initial population,
  • e is the base of the natural logarithm,
  • k is the growth rate constant.

Solving for variables in this equation, such as determining the time it takes for a population to double, involves using the techniques we've discussed for solving exponential equations.

Radioactive Decay

Another critical application is in the field of nuclear physics and chemistry, particularly in understanding radioactive decay. Radioactive substances decay exponentially, and the amount of a substance remaining after time t can be modeled by:

N(t) = N_0 * e^(-λt)

Where:

  • N(t) is the amount of the substance remaining at time t,
  • N_0 is the initial amount of the substance,
  • λ (lambda) is the decay constant.

This equation is used in carbon dating, medical imaging, and nuclear medicine to determine the age of artifacts, track the decay of radioactive isotopes, and more.

Compound Interest

In finance, exponential equations are used to calculate compound interest. The formula for compound interest is:

A = P(1 + r/n)^(nt)

Where:

  • A is the amount of money accumulated after n years, including interest,
  • P is the principal amount (the initial deposit),
  • r is the annual interest rate (as a decimal),
  • n is the number of times that interest is compounded per year,
  • t is the number of years the money is invested or borrowed for.

Solving for variables such as t (time to reach a certain amount) requires using logarithms, demonstrating another practical application of these mathematical skills.

Other Applications

Beyond these examples, exponential equations appear in various other contexts, including:

  • Electrical Engineering: Analyzing circuits with capacitors and inductors.
  • Chemical Kinetics: Studying reaction rates.
  • Computer Science: Analyzing algorithms and data structures.
  • Environmental Science: Modeling the spread of pollutants.

The ability to solve exponential equations provides a powerful tool for understanding and making predictions in these diverse fields.

In summary, solving the exponential equation e^(3x+2) = 1/e^4 involves rewriting the equation with the same base, equating the exponents, solving the resulting linear equation, and verifying the solution. The exact solution set for this equation is {-2}. We also explored an alternative method using logarithms, which is particularly useful for more complex exponential equations. The real-world applications of exponential equations, from population growth to radioactive decay and compound interest, highlight the importance of mastering these mathematical concepts. By understanding the properties of exponents and logarithms, we can effectively solve exponential equations and apply them to a wide range of practical problems. This article has provided a comprehensive guide to solving exponential equations, equipping you with the knowledge and skills to tackle similar problems with confidence.