Solving Exponential Equations A Step-by-Step Guide To Solving $e^x = E^{2x+5}$

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In the realm of mathematics, exponential equations hold a significant position, appearing in various fields such as calculus, differential equations, and mathematical modeling. Understanding how to solve these equations is crucial for students and professionals alike. In this article, we will delve into the step-by-step process of solving the exponential equation $e^x = e^{2x+5}$, providing a detailed explanation that caters to beginners while also offering insights for those more experienced in mathematics.

Understanding Exponential Equations

Before diving into the solution, it's essential to grasp the fundamentals of exponential equations. An exponential equation is one in which the variable appears in the exponent. These equations often involve constants, coefficients, and mathematical operations, but the defining characteristic is the presence of the variable in the exponent.

Exponential functions, in general, take the form $f(x) = a^x$, where $a$ is the base and $x$ is the exponent. The base $a$ is a constant, and the exponential function describes how the value changes as $x$ varies. In the given equation, $e^x = e^{2x+5}$, the base is the mathematical constant $e$, which is approximately equal to 2.71828. This constant, also known as Euler's number, is fundamental in calculus and many areas of mathematics. It arises naturally in the context of compound interest and continuous growth processes.

When dealing with exponential equations, a key property comes into play: if $a^m = a^n$, then $m = n$, provided that the base $a$ is the same and not equal to 1. This property forms the backbone of solving many exponential equations, as it allows us to equate the exponents and simplify the equation into a more manageable form. The exponential function is a one-to-one function, meaning that each input value corresponds to a unique output value, and vice versa. This property ensures that if two exponential expressions with the same base are equal, then their exponents must also be equal. Therefore, this characteristic enables us to solve exponential equations by focusing solely on the exponents.

Identifying Exponential Equations

Being able to identify exponential equations is the first step in solving them. Look for instances where the variable is part of the exponent. Common examples include equations like $2^x = 8$, $10^{2x-1} = 100$, and, of course, our equation, $e^x = e^{2x+5}$. The base of the exponential expression is also a crucial component to consider. The base can be any positive number except 1, as 1 raised to any power is always 1, which trivializes the equation. Recognizing the structure of an exponential equation is the foundation for applying the appropriate solving techniques. Exponential equations are prevalent in various mathematical contexts, including modeling population growth, radioactive decay, and compound interest. Therefore, mastering the techniques to solve these equations is an invaluable skill in mathematics and its applications.

Step-by-Step Solution of $e^x = e^{2x+5}$

Now, let's proceed with the step-by-step solution of the equation $e^x = e^{2x+5}$. This process demonstrates how to apply the fundamental properties of exponential functions to find the value of $x$ that satisfies the equation.

Step 1: Equate the Exponents

The cornerstone of solving exponential equations with the same base lies in the property that if $a^m = a^n$, then $m = n$. In our equation, $e^x = e^{2x+5}$, the base is $e$ on both sides. Therefore, we can equate the exponents:

$x = 2x + 5$

This step transforms the exponential equation into a simple linear equation, which is significantly easier to solve. The ability to equate exponents is a direct consequence of the one-to-one nature of exponential functions. It allows us to bypass the exponential nature of the equation and work with a more manageable algebraic expression. This property is essential for solving a wide range of exponential equations, making it a crucial tool in mathematical problem-solving.

Step 2: Solve the Linear Equation

Now we have a linear equation, $x = 2x + 5$. To solve for $x$, we need to isolate the variable on one side of the equation. We can start by subtracting $2x$ from both sides:

$x - 2x = 2x + 5 - 2x$

This simplifies to:

$-x = 5$

To find the value of $x$, we multiply both sides by -1:

$(-1)(-x) = (-1)(5)$

Which gives us:

$x = -5$

The solution to the linear equation is $x = -5$. This value represents the solution to the original exponential equation as well. Solving linear equations is a fundamental skill in algebra, and it often forms a critical step in solving more complex equations, including exponential equations. The process of isolating the variable involves performing algebraic operations on both sides of the equation to maintain equality while simplifying the equation until the variable is by itself.

Step 3: Verify the Solution

It's always a good practice to verify the solution by plugging it back into the original equation. In our case, the original equation is $e^x = e^{2x+5}$, and the solution we found is $x = -5$. Substituting $x = -5$ into the equation, we get:

$e^{-5} = e^{2(-5)+5}$

Simplifying the exponent on the right side:

$e^{-5} = e^{-10+5}$

$e^{-5} = e^{-5}$

Since both sides of the equation are equal, our solution $x = -5$ is correct. Verification is a crucial step in the problem-solving process, as it ensures that the solution satisfies the original equation and that no errors were made during the solving process. It provides confidence in the correctness of the answer and helps to avoid mistakes. In more complex equations, verification can be particularly important, as there may be extraneous solutions that do not satisfy the original equation.

Alternative Methods and Advanced Techniques

While the method described above is the most straightforward approach for solving the equation $e^x = e^{2x+5}$, it's worth noting that other techniques can be used, especially for more complex exponential equations. Furthermore, understanding advanced techniques can provide a broader perspective and enable you to tackle more challenging problems.

Using Logarithms

Logarithms are the inverse functions of exponentials and can be used to solve exponential equations. The natural logarithm, denoted as $\ln$, is the logarithm with base $e$. If we take the natural logarithm of both sides of the original equation, $e^x = e^{2x+5}$, we get:

$\ln(e^x) = \ln(e^{2x+5})$

Using the property that $\ln(e^u) = u$, we simplify the equation to:

$x = 2x + 5$

This is the same linear equation we obtained by equating the exponents directly. Thus, solving this equation will lead to the same solution, $x = -5$. The use of logarithms is particularly beneficial when the bases of the exponential expressions are different or when the equation cannot be easily simplified by equating exponents directly. Logarithms allow us to bring the exponents down and transform the exponential equation into a more manageable algebraic equation.

Dealing with Different Bases

When dealing with exponential equations that have different bases, such as $2^x = 3^{x+1}$, logarithms are essential. In such cases, equating the exponents directly is not possible. Instead, taking the logarithm of both sides allows us to isolate the variable $x$. For example, taking the natural logarithm of both sides of $2^x = 3^{x+1}$ gives:

$\ln(2^x) = \ln(3^{x+1})$

Using the power rule of logarithms, which states that $\ln(a^b) = b \ln(a)$, we get:

$x \ln(2) = (x+1) \ln(3)$

This equation can then be solved for $x$ using algebraic manipulation. Dealing with different bases often requires a deeper understanding of logarithmic properties and algebraic techniques. Logarithms provide a powerful tool for transforming exponential equations into forms that can be solved using standard algebraic methods.

Quadratic Forms in Exponential Equations

Some exponential equations can be transformed into quadratic equations through substitution. For instance, consider the equation $e^{2x} - 3e^x + 2 = 0$. This equation can be rewritten as $(e^x)^2 - 3e^x + 2 = 0$. By substituting $y = e^x$, we get a quadratic equation:

$y^2 - 3y + 2 = 0$

This quadratic equation can be factored as $(y-1)(y-2) = 0$, which gives us two possible values for $y$: $y = 1$ and $y = 2$. Substituting back $e^x$ for $y$, we get two exponential equations:

$e^x = 1$ and $e^x = 2$

The first equation, $e^x = 1$, gives us $x = 0$. The second equation, $e^x = 2$, gives us $x = \ln(2)$. This technique of transforming exponential equations into quadratic forms is a powerful method for solving equations that might otherwise seem intractable. It requires a keen eye for recognizing patterns and the ability to make appropriate substitutions.

Common Mistakes and Pitfalls

Solving exponential equations involves specific techniques, and it's easy to make mistakes if these techniques are not applied correctly. Recognizing common pitfalls can help prevent errors and ensure accurate solutions. Here are some common mistakes to watch out for:

Incorrectly Applying Logarithms

One common mistake is misapplying the properties of logarithms. For example, $\ln(a+b)$ is not equal to $\ln(a) + \ln(b)$. The correct property is $\ln(ab) = \ln(a) + \ln(b)$. Similarly, $\ln(a/b) = \ln(a) - \ln(b)$. When using logarithms to solve exponential equations, it's essential to apply these properties correctly to avoid errors. Understanding the fundamental properties of logarithms is crucial for their proper application. Misapplying these properties can lead to incorrect solutions, so careful attention to detail is necessary.

Forgetting to Verify Solutions

As mentioned earlier, verifying solutions is a crucial step in the problem-solving process. Sometimes, solutions obtained through algebraic manipulation may not satisfy the original equation due to extraneous roots. This is particularly true when dealing with logarithmic or radical equations. Always plug the solution back into the original equation to ensure it is valid. Verification provides a safeguard against errors and ensures the correctness of the solution.

Misunderstanding the Domain of Exponential Functions

Exponential functions are defined for all real numbers, but logarithmic functions are only defined for positive arguments. When solving exponential equations using logarithms, it's important to ensure that the arguments of the logarithms are positive. If a solution leads to a negative or zero argument in a logarithm, it is not a valid solution. Understanding the domain restrictions of logarithmic functions is essential for correctly solving exponential equations involving logarithms.

Errors in Algebraic Manipulation

Simple algebraic errors, such as incorrect distribution or combining like terms, can lead to incorrect solutions. Double-checking each step in the algebraic manipulation is essential to avoid these errors. Accuracy in algebraic manipulations is crucial for obtaining the correct solution. Even small errors can propagate through the steps and lead to a wrong answer.

Conclusion

In summary, solving the exponential equation $e^x = e^{2x+5}$ involves equating the exponents, solving the resulting linear equation, and verifying the solution. We found that $x = -5$ is the solution to the equation. Exponential equations are a fundamental concept in mathematics with broad applications in various fields. Mastering the techniques to solve them is an invaluable skill for anyone studying or working in mathematics, science, or engineering. By understanding the properties of exponential functions, applying the correct algebraic techniques, and avoiding common pitfalls, you can confidently solve a wide range of exponential equations. Continuous practice and attention to detail are key to mastering this important mathematical skill. This article has provided a comprehensive guide to solving exponential equations, from the basics to more advanced techniques. By following the steps outlined and understanding the underlying principles, you can enhance your problem-solving skills and gain a deeper appreciation for the beauty and power of mathematics.