Solving Exponential Equations 5e^(3x+2) = 6 A Step-by-Step Guide
Introduction
In the realm of mathematics, solving exponential equations is a fundamental skill with applications spanning various fields, including finance, physics, and engineering. Exponential equations, characterized by variables appearing in exponents, often require specific techniques to isolate the variable and determine its value. This comprehensive guide aims to elucidate the process of solving the exponential equation 5e^(3x+2) = 6 without the aid of a calculator, providing a step-by-step approach to understanding the underlying principles and arriving at the solution. Mastery of these techniques not only enhances mathematical proficiency but also equips individuals with the analytical skills necessary for tackling real-world problems involving exponential growth and decay.
Understanding Exponential Equations
Exponential equations are mathematical expressions where the variable appears in the exponent. These equations are prevalent in various scientific and mathematical contexts, modeling phenomena such as population growth, radioactive decay, and compound interest. Solving these equations requires a firm grasp of exponential properties and logarithmic functions. For instance, the equation 5e^(3x+2) = 6 is a classic example of an exponential equation where 'x' is the variable we aim to find. The base of the exponent is 'e,' the natural number approximately equal to 2.71828, making it a natural exponential equation. Understanding the behavior of exponential functions is crucial. Exponential functions either grow rapidly (when the base is greater than 1) or decay rapidly (when the base is between 0 and 1). The exponent determines the rate of this growth or decay, and manipulating the equation to isolate the variable in the exponent is the key to solving it. Recognizing the structure of exponential equations and their properties allows for a strategic approach to problem-solving, ensuring accurate and efficient solutions.
The Given Equation: 5e^(3x+2) = 6
The equation at hand, 5e^(3x+2) = 6, presents a typical scenario for solving exponential equations. The goal is to isolate 'x,' which is embedded within the exponent. This process involves a series of algebraic manipulations designed to peel away the layers surrounding 'x.' The initial step often involves isolating the exponential term, which in this case is e^(3x+2). Once the exponential term is isolated, the next crucial step is to apply the natural logarithm (ln) to both sides of the equation. The natural logarithm is the logarithm to the base 'e,' and it serves as the inverse function of the exponential function e^x. Applying the natural logarithm allows us to bring down the exponent, thereby freeing 'x' from its exponential position. This technique leverages the property of logarithms that ln(a^b) = b * ln(a). The subsequent steps involve basic algebraic manipulations such as division, subtraction, and division again to finally isolate 'x.' Each step must be performed meticulously to maintain the equality and ensure an accurate solution. Understanding the order of operations and the properties of logarithms is essential for successfully navigating through the solution process.
Step-by-Step Solution
Step 1: Isolate the Exponential Term
The first step in solving the equation 5e^(3x+2) = 6 is to isolate the exponential term, e^(3x+2). This is achieved by dividing both sides of the equation by 5. This operation ensures that the exponential term is by itself on one side of the equation, making it easier to apply further operations. The result of this step is:
e^(3x+2) = 6/5
Isolating the exponential term is a crucial step because it sets the stage for the application of logarithms, which is the key to bringing down the exponent and solving for 'x.' By performing this initial isolation, we simplify the equation and prepare it for the next phase of the solution process. This step underscores the importance of understanding basic algebraic manipulations and their role in simplifying complex equations.
Step 2: Apply the Natural Logarithm
With the exponential term isolated, the next crucial step is to apply the natural logarithm (ln) to both sides of the equation. The natural logarithm is the logarithm to the base 'e,' and it is the inverse function of the exponential function e^x. This step is pivotal because it allows us to bring down the exponent, effectively freeing 'x' from its exponential position. Applying the natural logarithm to both sides of the equation e^(3x+2) = 6/5 yields:
ln(e^(3x+2)) = ln(6/5)
Using the property of logarithms that ln(a^b) = b * ln(a), the left side of the equation simplifies to:
(3x+2) * ln(e) = ln(6/5)
Since ln(e) = 1, the equation further simplifies to:
3x + 2 = ln(6/5)
This transformation is a critical step in solving exponential equations, as it converts the exponential equation into a linear equation, which is much easier to solve. The application of the natural logarithm is a powerful technique that leverages the inverse relationship between exponential and logarithmic functions, providing a direct path to isolating the variable.
Step 3: Isolate the Variable x
After applying the natural logarithm and simplifying the equation to 3x + 2 = ln(6/5), the next step is to isolate the variable 'x.' This involves a series of algebraic manipulations to peel away the constants and coefficients surrounding 'x.' First, we subtract 2 from both sides of the equation to isolate the term containing 'x':
3x = ln(6/5) - 2
Next, we divide both sides of the equation by 3 to solve for 'x':
x = (ln(6/5) - 2) / 3
This is the exact solution for 'x.' It is expressed in terms of the natural logarithm, which is a common form for solutions to exponential equations. While this form may appear complex, it provides the precise value of 'x' that satisfies the original equation. If a numerical approximation is needed, a calculator could be used to evaluate ln(6/5), but the exact solution is often preferred for its accuracy and mathematical elegance. Isolating the variable 'x' demonstrates the power of algebraic manipulations in solving equations, highlighting the importance of performing operations in the correct order to achieve the desired result.
Final Answer
Expressing the Solution
The solution to the equation 5e^(3x+2) = 6 is x = (ln(6/5) - 2) / 3. This is the exact solution, expressed in terms of the natural logarithm. It represents the precise value of 'x' that satisfies the given equation. While a numerical approximation could be obtained using a calculator, the exact solution is often preferred for its accuracy and mathematical clarity. Expressing the solution in this form also allows for further mathematical analysis or manipulation if needed. For example, this solution could be used in subsequent calculations or compared with other solutions in a more complex problem. The ability to express solutions in exact forms is a hallmark of mathematical proficiency, demonstrating a deep understanding of the underlying principles and a commitment to precision.
Verifying the Solution
To ensure the accuracy of the solution x = (ln(6/5) - 2) / 3, it is essential to verify it by substituting it back into the original equation, 5e^(3x+2) = 6. This process confirms that the solution indeed satisfies the equation and that no errors were introduced during the solving process. Substituting the solution into the original equation, we get:
5e^(3((ln(6/5) - 2) / 3) + 2) = 6
Simplifying the exponent:
5e^(ln(6/5) - 2 + 2) = 6
5e^(ln(6/5)) = 6
Using the property that e^(ln(a)) = a, we have:
5 * (6/5) = 6
6 = 6
The equation holds true, confirming that the solution x = (ln(6/5) - 2) / 3 is correct. Verification is a critical step in problem-solving, as it provides assurance that the obtained solution is accurate and reliable. This practice reinforces the importance of precision and attention to detail in mathematical calculations. By verifying the solution, we not only validate our work but also deepen our understanding of the equation and the solution process.
Conclusion
Recap of the Solution Process
In conclusion, solving the exponential equation 5e^(3x+2) = 6 involved a series of carefully executed steps, each building upon the previous one to isolate the variable 'x.' The process began with isolating the exponential term by dividing both sides of the equation by 5. This prepared the equation for the application of the natural logarithm, which is the inverse function of the exponential function e^x. Applying the natural logarithm to both sides allowed us to bring down the exponent, transforming the exponential equation into a linear equation. The resulting equation, 3x + 2 = ln(6/5), was then solved for 'x' using basic algebraic manipulations, including subtraction and division. The final solution, x = (ln(6/5) - 2) / 3, represents the exact value of 'x' that satisfies the original equation. This step-by-step approach highlights the importance of understanding the properties of exponential and logarithmic functions, as well as the ability to apply algebraic techniques to solve equations. The solution process underscores the interconnectedness of mathematical concepts and the power of logical reasoning in problem-solving.
Importance of Mastering Exponential Equations
Mastering the techniques for solving exponential equations is crucial for several reasons. Exponential equations appear in a wide array of real-world applications, including population growth, radioactive decay, compound interest, and various scientific and engineering problems. A solid understanding of how to solve these equations equips individuals with the analytical skills necessary to model and solve these problems effectively. Furthermore, the skills developed in solving exponential equations, such as algebraic manipulation, applying inverse functions, and logical reasoning, are transferable to other areas of mathematics and problem-solving. These skills enhance mathematical proficiency and provide a foundation for more advanced topics. Additionally, the ability to solve exponential equations without the aid of a calculator fosters a deeper understanding of the underlying mathematical principles, promoting critical thinking and problem-solving abilities. In essence, mastering exponential equations is not just about solving specific problems; it's about developing a versatile and robust mathematical toolkit that can be applied across a wide range of contexts.
Final Answer
The final answer to the equation 5e^(3x+2) = 6 is:
x = (ln(6/5) - 2) / 3
