Solving Exponential Equations: A Step-by-Step Guide

Dec 5, 2025 by Alex Johnson 52 views

$Solving $7e^{10x} = 12$: A Comprehensive Guide$

Hey there, math enthusiasts! Today, we're going to dive into the world of solving exponential equations. Specifically, we'll tackle the equation $7e^{10x} = 12$ . Don't worry if exponential equations seem intimidating at first – we'll break down each step in a clear, easy-to-follow manner. By the end of this guide, you'll be able to confidently solve this type of problem and understand the underlying mathematical principles. Let's get started!

Understanding the Basics of Exponential Equations

Before we jump into the specific problem, let's brush up on what exponential equations are all about. Essentially, these equations involve a variable in the exponent. This variable is what we're trying to solve for. The key to solving these equations often involves using logarithms, which are the inverse of exponential functions. Remember that the exponential function $e^x$ is a fundamental concept in mathematics, appearing in various fields like calculus, physics, and finance. The number e, known as Euler's number, is approximately equal to 2.71828. Understanding the basics will make the process of solving $7e^{10x} = 12$ much smoother.

So, what are the primary strategies we'll be employing? First, our objective is to isolate the exponential term. That means getting the $e^{10x}$ part all by itself on one side of the equation. Once we've done that, we'll use the natural logarithm (ln), which is the logarithm with base e, to get rid of the exponent. This is where the inverse relationship between exponentials and logarithms becomes invaluable. The natural logarithm of $e^x$ is simply x. By applying ln to both sides of the equation, we can bring down the exponent and solve for our variable.

Why is understanding this essential? Because exponential equations are prevalent. They're not just theoretical exercises in textbooks; they have real-world applications. From calculating compound interest in investments to understanding population growth or radioactive decay, exponential functions are the mathematical backbone of many phenomena. Therefore, mastering the skills required to solve these equations gives you a significant advantage in various STEM fields and beyond. Grasping this also means understanding the properties of logarithms, such as the power rule (ln( $a^b$ ) = b ln(a)), which will be crucial for simplifying our expressions and arriving at the correct solution. Let’s get our hands dirty and start solving it.

Step-by-Step Solution of $7e^{10x} = 12$

Now, let's solve the equation $7e^{10x} = 12$ step-by-step. Remember, the goal is to isolate x. Here's the breakdown:

Isolate the Exponential Term: The first step is to isolate the exponential term, which is $e^{10x}$ . To do this, we need to get rid of the 7 that's multiplying it. We achieve this by dividing both sides of the equation by 7.

$7e^{10x} = 12$ $e^{10x} = rac{12}{7}$

Great, we've isolated the exponential part! Now, let's continue.
Take the Natural Logarithm of Both Sides: Since we have $e^{10x}$ , we'll take the natural logarithm (ln) of both sides. This is where the magic happens and we begin to untangle the exponent.

$ln(e^{10x}) = ln( rac{12}{7})$

By applying the natural logarithm, we are utilizing the inverse property, which is crucial for simplifying and solving the equation.
Use the Power Rule of Logarithms: The power rule of logarithms states that $ln(a^b) = b * ln(a)$ . In our case, this allows us to bring down the exponent 10x:

$10x * ln(e) = ln( rac{12}{7})$

Since $ln(e) = 1$ , the equation simplifies further.
Simplify: Now we simplify the equation, taking into account that $ln(e) = 1$ :

$10x * 1 = ln( rac{12}{7})$ $10x = ln( rac{12}{7})$
Solve for x: The last step involves solving for x. To do this, divide both sides of the equation by 10:

$x = rac{ln( rac{12}{7})}{10}$

Now we can calculate the final value by calculating the right side of the equation.
Calculate the Value: Use a calculator to find the natural logarithm of $rac{12}{7}$ and then divide by 10. Rounding to the nearest tenth as requested will get you the final solution.

$x hickapprox rac{0.5389965}{10}$ $x hickapprox 0.05389965$
Round the Answer: Round the answer to the nearest tenth, as requested. The final answer is:

$x hickapprox 0.1$

So, the solution to the equation $7e^{10x} = 12$ rounded to the nearest tenth is approximately 0.1. By following these steps, you've successfully solved an exponential equation!

Important Considerations and Tips

When working with exponential equations, there are a few important points to keep in mind to ensure accuracy and avoid common pitfalls. First, always double-check your work, particularly when using a calculator. Small errors can easily creep in when entering logarithmic values or performing calculations. Always isolate the exponential term first before taking the logarithm of both sides. This is a fundamental principle and is the most common mistake made by learners. Being methodical in this approach is essential for a correct solution.

Another important aspect is understanding the properties of logarithms. The power rule, as we’ve seen, is incredibly useful, but understanding the product and quotient rules will help you in more complex problems. These rules allow you to simplify and rewrite logarithmic expressions, which is useful when dealing with more complex scenarios. It's often helpful to keep these rules in a cheat sheet during your practice to refer to them, especially when you are just beginning to master these concepts. Also, be mindful of the domain of logarithmic functions. You can only take the logarithm of a positive number. If, in your calculations, you encounter a situation where you need to take the logarithm of a negative number or zero, something is wrong, and you should re-evaluate your steps.

Finally, practicing a variety of problems is the key to mastering these concepts. Start with simpler problems and gradually move to more complex ones. The more you work with these equations, the more familiar you will become with the techniques involved. Look for problems where the exponential term is more complex, such as those with quadratic equations in the exponent, to challenge yourself. Solving these problems will help solidify your understanding and increase your confidence. Consistency is key when dealing with exponential equations; regular practice makes perfect, and you'll find yourself able to solve complex equations with ease.

Further Practice and Resources

If you want to delve deeper, there are several resources available to enhance your understanding. Many online platforms offer practice problems, step-by-step video tutorials, and interactive exercises. Check out Khan Academy, which provides free, comprehensive lessons on exponential functions and logarithms. Other great websites include Mathway and Wolfram Alpha, where you can input problems and receive detailed solutions. These platforms can be a great place to check your work or understand how to solve more complex problems.

Textbooks and study guides for algebra and precalculus are also useful resources. They provide detailed explanations and practice problems to hone your skills. Look for books that cover exponential and logarithmic functions and provide ample practice examples. Working through these resources will help reinforce the concepts and provide additional practice, ensuring you have a strong grasp of the subject.

Moreover, don't hesitate to seek help from your teachers, tutors, or online forums if you struggle with any concept. Discussing the problems and concepts with others can improve understanding. Often, clarifying a tricky point with a peer or expert can make the difference between a vague understanding and solid comprehension. Moreover, group study is extremely effective as it allows you to learn from your peers, discover alternate solution paths, and reinforce your own knowledge by explaining it to others.

Conclusion

Congratulations! You've successfully solved the exponential equation $7e^{10x} = 12$ . We've walked through the key steps: isolating the exponential term, taking the natural logarithm, simplifying using logarithmic properties, solving for x, and rounding the answer. With consistent practice and the right resources, you'll be able to solve these types of equations with confidence. Remember that solving exponential equations is a valuable skill, with applications in various fields of science and mathematics. Keep practicing, and you'll become more proficient over time!

For more detailed information and practice problems, you can visit Khan Academy https://www.khanacademy.org/math/algebra2/x2ec2f6f8303f9052:logs for a comprehensive lesson on logarithms and solving exponential equations.