Solving The Exponential Equation E^(-5x)=(e^6)^(x+11)

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Understanding Exponential Equations

Before we jump into the solution, let's briefly discuss exponential equations. An exponential equation is an equation in which the variable appears in the exponent. These types of equations often arise in various fields, including finance, physics, and biology, where they are used to model phenomena like compound interest, radioactive decay, and population growth. The key to solving exponential equations lies in manipulating the equation to isolate the variable, and one common technique involves using logarithms or the properties of exponents.

In our case, we have the equation $e^{-5x} = (e^6)^{x+11}$. This equation features the exponential constant 'e', which is approximately 2.71828. The presence of 'e' suggests that logarithms might be a useful tool in solving this equation. However, before we reach for logarithms, let's see if we can simplify the equation using the properties of exponents.

Step 1: Simplify the Equation Using Exponent Rules

The first step in solving the equation is to simplify it using the properties of exponents. Specifically, we'll use the power of a power rule, which states that $(a^m)^n = a^{mn}$. Applying this rule to the right side of the equation, $(e^6)^{x+11}$, we get:

$(e^6)^{x+11} = e^{6(x+11)}$

Now, we need to distribute the 6 in the exponent:

$e^{6(x+11)} = e^{6x + 66}$

So, our equation now looks like this:

$e^{-5x} = e^{6x + 66}$

This simplification is crucial because it allows us to directly compare the exponents on both sides of the equation. When the bases are the same, as they are in this case (both sides have a base of e), we can equate the exponents. This is a fundamental property of exponential equations and is key to finding our solution.

Step 2: Equate the Exponents

Since the bases are the same, we can now equate the exponents. This means we set the exponent on the left side of the equation equal to the exponent on the right side:

$-5x = 6x + 66$

We've now transformed our exponential equation into a simple linear equation. This is a significant step because linear equations are much easier to solve. The next step involves isolating the variable x on one side of the equation.

Step 3: Solve for x

To solve for x, we need to isolate it on one side of the equation. Let's start by adding 5x to both sides of the equation:

$-5x + 5x = 6x + 66 + 5x$

This simplifies to:

$0 = 11x + 66$

Next, subtract 66 from both sides:

$0 - 66 = 11x + 66 - 66$

$-66 = 11x$

Finally, divide both sides by 11:

$-66 / 11 = 11x / 11$

$x = -6$

So, the solution to the equation is x = -6. This means that when x is -6, the equation $e^{-5x} = (e^6)^{x+11}$ holds true. We have successfully found the value of x that satisfies the equation.

Step 4: Verify the Solution

It's always a good practice to verify your solution by plugging it back into the original equation. This helps ensure that you haven't made any errors in your calculations. Let's substitute x = -6 into the original equation:

$e^{-5(-6)} = (e^6)^{-6+11}$

Simplify the exponents:

$e^{30} = (e^6)^{5}$

Apply the power of a power rule again:

$e^{30} = e^{6 * 5}$

$e^{30} = e^{30}$

Since both sides of the equation are equal, our solution x = -6 is correct. This verification step confirms that we have found the accurate value of x.

Conclusion

In this article, we have successfully solved the exponential equation $e^{-5x} = (e^6)^{x+11}$. We achieved this by following a systematic approach that involved simplifying the equation using exponent rules, equating the exponents, solving the resulting linear equation, and verifying the solution. The value of x that satisfies the equation is -6. By understanding these steps, you can confidently solve similar exponential equations.

Solving exponential equations requires a solid understanding of exponent rules and algebraic manipulation. By breaking down the problem into smaller, manageable steps, we can arrive at the solution. Remember to always verify your solution to ensure accuracy. This step-by-step approach is applicable to a wide range of exponential equations, making it a valuable tool in your mathematical toolkit.

While the method we used to solve this specific equation is effective, there are other strategies and considerations that can be helpful when dealing with exponential equations in general. Here are some additional tips to keep in mind:

• Logarithms: When the bases of the exponents cannot be easily made the same, logarithms are your best friend. Taking the logarithm of both sides of an exponential equation can help you bring the variable down from the exponent. The natural logarithm (ln) is particularly useful when dealing with the exponential constant e.
• Rewrite with the Same Base: If possible, try to rewrite both sides of the equation with the same base. This allows you to equate the exponents, as we did in our example. Look for common factors or powers that might help you achieve this.
• Substitution: For more complex equations, substitution can be a powerful technique. If you see a repeating exponential term, try substituting a variable for that term. This can transform the equation into a more familiar form, such as a quadratic equation.
• Check for Extraneous Solutions: When using logarithms, it's essential to check for extraneous solutions. Logarithms are only defined for positive arguments, so any solution that results in taking the logarithm of a negative number or zero is not valid.
• Graphical Solutions: Sometimes, an approximate solution can be found graphically. By plotting both sides of the equation as separate functions, the intersection points represent the solutions to the equation.
• Practice, Practice, Practice: The best way to become comfortable with solving exponential equations is to practice. Work through a variety of problems, and don't be afraid to make mistakes – they are valuable learning opportunities.

By mastering these techniques and practicing regularly, you'll develop the skills needed to tackle a wide range of exponential equations with confidence. Exponential equations are a fundamental concept in mathematics, and their applications extend far beyond the classroom.

Exponential equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding these applications can help you appreciate the importance of mastering exponential equations.

• Finance: Compound interest is a classic example of exponential growth. The amount of money you earn on an investment grows exponentially over time, thanks to the power of compounding. Exponential equations are used to calculate future values, interest rates, and investment time horizons.
• Population Growth: Population growth often follows an exponential pattern. The number of individuals in a population can increase rapidly, especially when resources are abundant. Exponential models are used to predict population sizes and growth rates.
• Radioactive Decay: Radioactive isotopes decay exponentially over time. The half-life of an isotope is the time it takes for half of the material to decay. Exponential equations are used to determine the age of artifacts and geological samples through carbon dating and other radiometric dating methods.
• Medicine: Drug concentration in the body decays exponentially as the drug is metabolized and eliminated. Exponential equations are used to determine drug dosages and dosing intervals to maintain therapeutic levels.
• Physics: Many physical phenomena, such as the discharge of a capacitor or the cooling of an object, follow exponential patterns. Exponential equations are used to model these processes and make predictions about their behavior.
• Computer Science: Exponential functions play a role in algorithms and data structures. For example, the time complexity of some algorithms can be exponential, meaning the time required to run the algorithm increases exponentially with the size of the input.

These are just a few examples of the many real-world applications of exponential equations. By understanding these applications, you can see how exponential equations are used to solve practical problems and make informed decisions in various fields.

In conclusion, solving exponential equations is a valuable skill that has numerous applications in the real world. By following a systematic approach and practicing regularly, you can master this skill and apply it to solve a wide range of problems. Remember to break down complex equations into smaller steps, utilize exponent rules and logarithms when appropriate, and always verify your solutions. With dedication and practice, you can become proficient in solving exponential equations and unlock their power in various fields.

Keywords

exponential equations, solve for x, exponent rules, logarithms, verification, real-world applications, equation, mathematics

Article Summary

This article provides a comprehensive guide to solving the exponential equation $e^{-5x} = (e^6)^{x+11}$. It begins by explaining the concept of exponential equations and the importance of mastering them. The article then presents a step-by-step solution to the equation, which includes simplifying the equation using exponent rules, equating the exponents, solving the resulting linear equation, and verifying the solution. The article also offers additional tips for solving exponential equations, such as using logarithms, rewriting with the same base, and checking for extraneous solutions. Furthermore, it highlights the numerous real-world applications of exponential equations in fields such as finance, population growth, radioactive decay, medicine, physics, and computer science. The article concludes by emphasizing the value of solving exponential equations and encouraging readers to practice regularly to improve their skills.

Solve the equation. $e^{-5 x}=\left(e^{6}\right)^{x+11}$

$x=$ (Type an integer or a simplified fraction.)

To solve the exponential equation $e^{-5x} = (e^6)^{x+11}$, we'll break it down step by step. Our primary goal is to isolate x. We'll start by simplifying the equation using the properties of exponents, then we'll solve for x algebraically. Remember, understanding exponential equations and their properties is crucial for various mathematical applications. So, let's dive in and solve this equation together.

Step 1: Simplify Using Exponent Rules

The given equation is: $e^{-5x} = (e^6)^{x+11}$.

The first thing we need to do is simplify the right side of the equation. We can use the power of a power rule, which states that $(a^m)^n = a^{mn}$. Applying this rule to our equation, we get:

$(e^6)^{x+11} = e^{6(x+11)}$

Now, we need to distribute the 6 in the exponent:

$e^{6(x+11)} = e^{6x + 66}$

So, our equation now looks like this:

$e^{-5x} = e^{6x + 66}$

Step 2: Equate the Exponents

Now that we have the same base (e) on both sides of the equation, we can equate the exponents. This is a crucial step because it transforms the exponential equation into a simpler linear equation. When the bases are equal, the exponents must also be equal for the equation to hold true. Therefore, we can set the exponents equal to each other:

$-5x = 6x + 66$

Step 3: Solve the Linear Equation

We now have a linear equation that we can solve for x. Our goal is to isolate x on one side of the equation. Let's start by getting all the x terms on one side. Add 5x to both sides of the equation:

$-5x + 5x = 6x + 66 + 5x$

This simplifies to:

$0 = 11x + 66$

Next, we want to isolate the term with x. Subtract 66 from both sides:

$0 - 66 = 11x + 66 - 66$

$-66 = 11x$

Finally, divide both sides by 11 to solve for x:

$\frac{-66}{11} = \frac{11x}{11}$

$x = -6$

So, the solution to the equation is x = -6.

Step 4: Verify the Solution

To ensure our solution is correct, we should always verify it by plugging it back into the original equation. Let's substitute x = -6 into the original equation:

$e^{-5(-6)} = (e^6)^{-6+11}$

Simplify the exponents:

$e^{30} = (e^6)^{5}$

Apply the power of a power rule again:

$e^{30} = e^{6 * 5}$

$e^{30} = e^{30}$

Since both sides of the equation are equal, our solution x = -6 is correct. This verification step is essential to avoid errors and ensures that we have found the accurate value of x.

Conclusion

In this article, we have successfully solved the exponential equation $e^{-5x} = (e^6)^{x+11}$. We followed a systematic approach that involved simplifying the equation using exponent rules, equating the exponents, solving the resulting linear equation, and verifying the solution. The value of x that satisfies the equation is -6. Understanding these steps will help you tackle similar exponential equations with confidence. Remember, practice is key to mastering mathematical concepts. By working through various problems, you will become more comfortable and proficient in solving exponential equations.

Solving exponential equations is a fundamental skill in mathematics with applications in various fields, including finance, physics, and engineering. The ability to manipulate exponents and solve for unknowns is crucial for understanding and modeling real-world phenomena. By mastering these techniques, you can unlock a deeper understanding of the mathematical principles that govern our world.

In conclusion, the solution to the equation $e^{-5x} = (e^6)^{x+11}$ is x = -6. We arrived at this solution by applying the power of a power rule, equating exponents, solving the resulting linear equation, and verifying our answer. This step-by-step approach provides a clear and concise method for solving similar exponential equations. Remember to always verify your solutions to ensure accuracy and deepen your understanding of the concepts involved. Practice makes perfect, so keep solving and exploring the world of exponential equations!

Keywords

Solve the equation, exponential equation, exponent rules, equate the exponents, linear equation, simplify, power of a power, verify the solution, mathematics, step-by-step guide, x = -6