Solve $e^{x+3} = 40^x$: Exponential Equation Guide

Aug 7, 2025 by Mei Lin 51 views

$Solving Exponential Equations: A Step-by-Step Guide to $e^{x+3}=40^x$$

Hey guys! Let's dive into solving exponential equations, specifically the equation $e^{x+3} = 40^x$ . Exponential equations can seem tricky at first, but with a systematic approach, we can conquer them. In this guide, we'll break down each step, making sure you understand the logic behind it. So, grab your pencils, and let's get started!

Understanding Exponential Equations

Before we jump into solving, let's quickly recap what exponential equations are. An exponential equation is one in which the variable appears in the exponent. For instance, $2^x = 8$ and our equation, $e^{x+3} = 40^x$ , are both exponential equations. The key to solving these equations often lies in manipulating them so that we can isolate the variable. One common technique involves using logarithms, which are the inverse operations of exponentiation. Logarithms allow us to "bring down" the exponent, making it easier to solve for the variable.

The natural logarithm, denoted as ln, is particularly useful when dealing with the base e (Euler's number, approximately 2.71828). Remember that $ln(e^x) = x$ , which is a fundamental property we'll use later. For other bases, we can use the common logarithm (base 10) or any other logarithm, but the natural logarithm is often the most convenient when e is involved. Now that we have a solid understanding of exponential equations and the role of logarithms, let's get back to our specific problem.

Breaking Down the Equation $e^{x+3} = 40^x$

Our mission is to solve for x in the equation $e^{x+3} = 40^x$ . The first step in tackling this equation is to apply a logarithm to both sides. This will help us untangle the exponents. Since we have e on one side, the natural logarithm (ln) is our best friend here. Applying the natural logarithm to both sides gives us:

$ln(e^{x+3}) = ln(40^x)$

Now, we use the power rule of logarithms, which states that $ln(a^b) = b * ln(a)$ . Applying this rule to both sides, we get:

$(x+3) * ln(e) = x * ln(40)$

Remember that $ln(e) = 1$ , so the left side simplifies beautifully to:

$x + 3 = x * ln(40)$

We're getting closer! Our next goal is to isolate x. To do this, we need to get all the terms involving x on one side of the equation and the constants on the other side. Let's subtract x * ln(40)* from both sides:

$x + 3 - x * ln(40) = 0$

Now, subtract 3 from both sides:

$x - x * ln(40) = -3$

Isolating and Solving for x

We're on the home stretch! Now, we need to factor out x from the left side of the equation. This is a crucial step in isolating x. Factoring out x gives us:

$x(1 - ln(40)) = -3$

Now, we can finally isolate x by dividing both sides by $(1 - ln(40))$ :

$x = \frac{-3}{1 - ln(40)}$

This is our solution! However, to make it more understandable, we can calculate the value of $ln(40)$ and simplify further. The natural logarithm of 40 is approximately 3.688879. Plugging this value into our equation, we get:

$x = \frac{-3}{1 - 3.688879}$

$x = \frac{-3}{-2.688879}$

$x ≈ 1.1157$

So, the solution to the equation $e^{x+3} = 40^x$ is approximately x = 1.1157. Remember, it's always a good idea to plug your solution back into the original equation to check if it's correct. Let's move on to verifying our solution.

Verifying the Solution

To ensure our solution is correct, we'll substitute x ≈ 1.1157 back into the original equation $e^{x+3} = 40^x$ . This step is crucial in confirming that our calculations are accurate and that we haven't made any mistakes along the way. Plugging in the value, we get:

$e^{1.1157+3} = 40^{1.1157}$

Let's simplify both sides of the equation:

$e^{4.1157} ≈ 40^{1.1157}$

Using a calculator, we find that:

$e^{4.1157} ≈ 61.313$

and

$40^{1.1157} ≈ 61.310$

The values are very close, and the slight difference is due to rounding. This confirms that our solution, x ≈ 1.1157, is indeed correct. Verification is an essential step in solving equations, as it helps catch any errors that might have occurred during the process. Now that we've verified our solution, let's summarize the steps we took to solve this exponential equation.

Summary of Steps to Solve $e^{x+3} = 40^x$

Let's recap the steps we took to solve the exponential equation $e^{x+3} = 40^x$ . This will not only reinforce your understanding but also provide a clear roadmap for tackling similar problems in the future.

Apply the Natural Logarithm: Take the natural logarithm of both sides of the equation. This is a crucial step because it allows us to use the properties of logarithms to simplify the equation.

$ln(e^{x+3}) = ln(40^x)$
Use the Power Rule of Logarithms: Apply the power rule, which states that $ln(a^b) = b * ln(a)$ , to both sides. This brings the exponents down as coefficients.

$(x+3) * ln(e) = x * ln(40)$
Simplify: Since $ln(e) = 1$ , the equation simplifies to:

$x + 3 = x * ln(40)$
Rearrange the Equation: Get all terms involving x on one side and constants on the other side. This is a standard algebraic manipulation to isolate the variable.

$x - x * ln(40) = -3$
Factor out x: Factor x from the terms on one side of the equation.

$x(1 - ln(40)) = -3$
Isolate x: Divide both sides by $(1 - ln(40))$ to solve for x.

$x = \frac{-3}{1 - ln(40)}$
Calculate and Approximate: Calculate the value of $ln(40)$ and approximate the solution.

$x ≈ 1.1157$
Verify the Solution: Substitute the value of x back into the original equation to ensure it is correct. This step is essential to catch any potential errors.

By following these steps, you can confidently solve a wide range of exponential equations. Remember, practice makes perfect, so the more you solve, the more comfortable you'll become with these techniques. Now, let's discuss some common mistakes to avoid when solving exponential equations.

Common Mistakes to Avoid

When solving exponential equations, there are a few common pitfalls that students often encounter. Being aware of these mistakes can help you avoid them and ensure you arrive at the correct solution. Let's highlight some of these common errors:

Incorrectly Applying Logarithm Properties: One of the most frequent mistakes is misapplying the properties of logarithms. For example, students might incorrectly assume that $ln(a + b) = ln(a) + ln(b)$ , which is not true. The correct property is $ln(a * b) = ln(a) + ln(b)$ . Make sure you have a solid grasp of logarithm properties before tackling exponential equations.
Forgetting to Apply the Logarithm to All Terms: When taking the logarithm of both sides of an equation, it's crucial to apply it to every term. For instance, in the equation $e^{x+3} = 40^x$ , you must take the logarithm of both $e^{x+3}$ and $40^x$ . Skipping this step can lead to an incorrect solution.
Misunderstanding the Order of Operations: Exponential equations often involve multiple operations, and it's essential to follow the correct order of operations (PEMDAS/BODMAS). Make sure to address exponents before multiplication or division and addition or subtraction.
Rounding Errors: Rounding intermediate values too early can introduce errors in your final answer. It's best to keep as many decimal places as possible during calculations and only round the final answer to the required precision.
Not Verifying the Solution: As we emphasized earlier, verifying your solution by plugging it back into the original equation is a critical step. Failing to do so means you might not catch errors that occurred during the solving process.
Incorrectly Factoring: Factoring is a common technique in solving equations, but it's also a place where mistakes can happen. Double-check your factoring steps to ensure you've done it correctly. In our example, factoring out x correctly is crucial for isolating the variable.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving exponential equations. Now, let's look at some additional practice problems to further hone your skills.

Additional Practice Problems

To solidify your understanding of solving exponential equations, let's work through a few more examples. Practice is key to mastering any mathematical concept, and these problems will give you the opportunity to apply the techniques we've discussed.

Solve for x: $3^{2x-1} = 81$
Solve for x: $5^{x+2} = 25^x$
Solve for x: $2e^{3x} = 16$
Solve for x: $7^{x-1} = 49^{x+1}$
Solve for x: $e^{2x} - 5e^x + 6 = 0$

These problems offer a range of challenges and will help you practice different scenarios you might encounter when solving exponential equations. Remember to apply the steps we've discussed, including taking logarithms, using the properties of logarithms, and verifying your solutions.

Feel free to work through these problems on your own, and if you get stuck, revisit the steps and explanations we've covered in this guide. The more you practice, the more comfortable and confident you'll become in solving exponential equations. Keep up the great work!

Solving exponential equations can seem daunting at first, but with a clear understanding of the steps involved and consistent practice, you can master this skill. Remember to take it one step at a time, and don't hesitate to review the concepts and techniques we've covered in this guide. Happy solving, guys!