Solving Exponential Equations A Step-by-Step Guide To 5(2^(x+4)) = 15
In this article, we will delve into the process of solving the exponential equation 5(2^(x+4)) = 15. Exponential equations, characterized by the presence of a variable in the exponent, are a fundamental concept in mathematics with applications spanning various fields, including finance, physics, and computer science. Mastering the techniques for solving these equations is crucial for anyone pursuing studies or careers in these areas. This guide will provide a step-by-step solution, ensuring clarity and understanding at each stage. We will cover the initial simplification, the application of logarithms to isolate the variable, and the final calculation of the solution, rounded to the nearest thousandth. Additionally, we will explore the underlying principles of exponential functions and logarithms, reinforcing the theoretical foundation necessary for tackling more complex problems. Understanding these concepts will not only help you solve this particular equation but also equip you with the skills to approach a wide range of mathematical challenges. Let's embark on this journey of mathematical exploration and unravel the intricacies of exponential equations together.
Understanding Exponential Equations
Exponential equations, at their core, involve a variable in the exponent, making them distinct from polynomial equations where the variable resides in the base. The general form of an exponential equation is a^x = b, where 'a' is the base, 'x' is the exponent, and 'b' is the result. The key to solving these equations lies in understanding the inverse relationship between exponential functions and logarithms. Logarithms provide a way to "undo" the exponentiation, allowing us to isolate the variable. Before diving into the specific steps for solving 5(2^(x+4)) = 15, it's important to grasp the fundamental properties of exponents and logarithms. For example, the property a^(m+n) = a^m * a^n is often used to simplify exponential expressions, while the logarithmic property log_a(b^c) = c * log_a(b) is crucial for bringing the exponent down. These properties are the cornerstones of solving exponential equations, enabling us to manipulate the equation into a form where the variable can be easily isolated. Without a solid understanding of these principles, the process of solving exponential equations can seem like a series of arbitrary steps. However, with a clear grasp of the underlying theory, the solution becomes a logical and intuitive process. Furthermore, understanding the nature of exponential growth and decay is vital for interpreting the solutions in real-world contexts. Exponential functions model phenomena that increase or decrease at a rate proportional to their current value, making them applicable to various situations, from population growth to radioactive decay.
Step-by-Step Solution of 5(2^(x+4)) = 15
To solve the exponential equation 5(2^(x+4)) = 15, we will follow a systematic approach that involves simplifying the equation, applying logarithms, and isolating the variable 'x'. The first step is to isolate the exponential term, 2^(x+4). This can be achieved by dividing both sides of the equation by 5. This isolates the exponential term on one side, making it easier to apply logarithmic properties in the subsequent steps. After this initial simplification, we are left with a simpler exponential equation that can be further manipulated using logarithms. The next step involves taking the logarithm of both sides of the equation. The choice of the base of the logarithm is arbitrary, but using the common logarithm (base 10) or the natural logarithm (base e) is often convenient because these logarithms are available on most calculators. Applying the logarithm allows us to bring the exponent down using the property log_a(b^c) = c * log_a(b). This is a crucial step in solving exponential equations because it transforms the problem from an exponential one to a linear one, which is much easier to solve. Once the exponent is brought down, we can isolate 'x' using basic algebraic manipulations. This involves distributing, combining like terms, and performing inverse operations to get 'x' by itself. Finally, after isolating 'x', we can calculate its value using a calculator. The solution is often an irrational number, so it is typically rounded to a certain number of decimal places, as specified in the problem. In this case, we are asked to round to the nearest thousandth, so we will round our final answer to three decimal places.
Detailed Steps to Solve the Equation
Let's break down the solution to 5(2^(x+4)) = 15 step-by-step.
- Isolate the exponential term: Divide both sides of the equation by 5:
This step simplifies the equation by isolating the exponential expression, making it ready for the next phase of solving.5(2^(x+4)) / 5 = 15 / 5 2^(x+4) = 3 - Apply logarithms to both sides: Take the logarithm of both sides of the equation. We can use the common logarithm (log base 10) or the natural logarithm (log base e). For this example, let's use the natural logarithm (ln):
Applying logarithms is essential for dealing with exponential equations, as it allows us to utilize logarithmic properties to bring the exponent down.ln(2^(x+4)) = ln(3) - Use the logarithm power rule: Apply the power rule of logarithms, which states that ln(a^b) = b * ln(a):
This rule is crucial for transforming the exponential equation into a linear one, making it solvable using basic algebraic techniques.(x+4) * ln(2) = ln(3) - Isolate (x+4): Divide both sides by ln(2):
By isolating (x+4), we move closer to solving for 'x'. This step involves applying the inverse operation of multiplication, which is division.(x+4) = ln(3) / ln(2) - Isolate x: Subtract 4 from both sides:
Finally, isolating 'x' allows us to determine its value. This is achieved by subtracting 4 from both sides of the equation.x = (ln(3) / ln(2)) - 4 - Calculate the value of x: Use a calculator to find the numerical value of x:
This final step involves using a calculator to obtain a numerical approximation of the solution. Logarithmic and exponential values are often irrational, so a calculator is necessary to get a decimal approximation.x ≈ (1.0986 / 0.6931) - 4 x ≈ 1.5850 - 4 x ≈ -2.415 - Round to the nearest thousandth: Round the value of x to the nearest thousandth:
Rounding to the nearest thousandth is a common practice to provide a precise but practical solution. It involves looking at the digit in the ten-thousandths place to determine whether to round up or down.x ≈ -2.415
Therefore, the solution to the equation 5(2^(x+4)) = 15, rounded to the nearest thousandth, is approximately -2.415. This detailed breakdown of the steps not only provides the solution but also emphasizes the underlying principles and techniques involved in solving exponential equations. Understanding these principles is essential for tackling more complex problems and applying these concepts in various fields.
Verification of the Solution
To ensure the accuracy of our solution, it is crucial to verify it by substituting the calculated value of x back into the original equation, 5(2^(x+4)) = 15. This process involves replacing 'x' with our solution, approximately -2.415, and evaluating the expression to see if it equals 15. The verification step serves as a safeguard against errors that may have occurred during the solving process, such as incorrect application of logarithmic properties or arithmetic mistakes. By substituting the value back into the original equation, we can confirm whether our solution satisfies the equation's conditions. If the left-hand side of the equation equals the right-hand side after the substitution, then our solution is verified. If not, it indicates that there may be an error in our calculations, and we need to review the steps to identify and correct the mistake. Verification is a fundamental practice in mathematics, particularly when dealing with complex equations, as it provides confidence in the correctness of the solution. It also reinforces the understanding of the relationship between the equation and its solution. Furthermore, verification can be a valuable learning tool, as it helps to identify areas of confusion or misunderstanding in the solving process. Therefore, always make it a habit to verify your solutions, especially in examinations or when applying these concepts in real-world scenarios. This meticulous approach not only ensures accuracy but also enhances your problem-solving skills and mathematical intuition.
Conclusion
In conclusion, we have successfully solved the exponential equation 5(2^(x+4)) = 15 by systematically applying the principles of logarithms and algebraic manipulation. The solution, rounded to the nearest thousandth, is approximately -2.415. This process involved several key steps, including isolating the exponential term, applying logarithms to both sides, utilizing the power rule of logarithms, isolating the variable 'x', and finally, calculating and rounding the numerical value. Each step was carefully explained to provide a clear understanding of the underlying concepts and techniques. Furthermore, we emphasized the importance of verifying the solution by substituting it back into the original equation, ensuring accuracy and reinforcing the understanding of the relationship between the equation and its solution. Mastering the techniques for solving exponential equations is crucial for various mathematical applications and real-world problems. Exponential functions and logarithms are fundamental concepts in mathematics, with applications spanning finance, physics, computer science, and many other fields. By understanding the properties of exponents and logarithms, and by practicing the steps involved in solving exponential equations, you can develop the skills and confidence to tackle more complex mathematical challenges. This comprehensive guide not only provides a solution to a specific problem but also aims to enhance your overall mathematical understanding and problem-solving abilities. Remember, practice is key to mastering these concepts, so continue to explore different types of exponential equations and apply the techniques learned in this article.
