Solving 4^(x-2) = 7 Exact And Approximate Solutions

Jul 16, 2025 by ADMIN 52 views

Solving the Exponential Equation 4^(x-2) = 7: An Exact and Approximate Solution

Introduction

In this article, we will delve into solving the exponential equation 4^(x-2) = 7. Exponential equations like this one appear frequently in various fields, such as finance, physics, and engineering. Understanding how to solve them is a crucial skill. Our approach will involve finding both an exact solution using the natural logarithm (ln) function and an approximate solution rounded to four decimal places. This comprehensive guide will walk you through each step, ensuring clarity and understanding. The solution will involve manipulating the equation using logarithmic properties to isolate x. We'll also explore the importance of understanding logarithmic functions in solving exponential equations. The use of the natural logarithm will provide us with a precise solution, while the decimal approximation gives a practical understanding of the value.

Understanding Exponential Equations

To begin, let's understand exponential equations generally. These equations involve a variable in the exponent, such as in our example, 4^(x-2) = 7. To solve these equations, we often use logarithms, which are the inverse operations of exponentiation. The logarithm of a number y with respect to a base b is the exponent to which b must be raised to produce y. In mathematical terms, if b^x = y, then log_b(y) = x. When dealing with exponential equations, it's crucial to remember the properties of exponents and logarithms. These properties allow us to manipulate the equations and isolate the variable. One of the most commonly used properties is the power rule of logarithms, which states that log_b(a^c) = c * log_b(a). This rule is particularly useful when the variable is in the exponent. Another important aspect of exponential equations is understanding the behavior of exponential functions. These functions either increase rapidly (if the base is greater than 1) or decrease rapidly (if the base is between 0 and 1) as the exponent changes. This behavior is important in applications such as population growth and radioactive decay. We will use these concepts to solve our specific equation.

Using the Natural Logarithm (ln)

The natural logarithm (ln) is the logarithm to the base e, where e is an irrational constant approximately equal to 2.71828. The natural logarithm is particularly useful in calculus and other advanced mathematical applications, but it is also incredibly helpful in solving exponential equations. The primary reason for this is that the natural logarithm has a direct relationship with the exponential function e^x. Specifically, ln(e^x) = x and e^(ln(x)) = x. This inverse relationship makes the natural logarithm a powerful tool for simplifying equations involving exponential terms. When solving exponential equations, the goal is to isolate the variable, which is often in the exponent. Taking the natural logarithm of both sides of an equation allows us to bring the exponent down as a coefficient, making it easier to solve for the variable. Furthermore, the properties of logarithms allow us to simplify complex expressions, turning multiplication inside the logarithm into addition outside the logarithm, and vice versa. This flexibility makes the natural logarithm a versatile tool in mathematical problem-solving. We will now apply the natural logarithm to solve the equation 4^(x-2) = 7.

Solving 4^(x-2) = 7: Exact Solution

Here, we will discuss the process of finding the exact solution for the equation 4^(x-2) = 7. To achieve this, we'll utilize the natural logarithm (ln). The initial step is to apply the natural logarithm to both sides of the equation. This maintains the equality while allowing us to use logarithmic properties to simplify the equation. By taking the natural logarithm of both sides, we get ln(4^(x-2)) = ln(7). Next, we apply the power rule of logarithms, which states that ln(a^b) = b * ln(a). Applying this rule, we rewrite the left side of the equation as (x-2) * ln(4) = ln(7). The next step involves isolating the term (x-2). We do this by dividing both sides of the equation by ln(4), resulting in x - 2 = ln(7) / ln(4). Now, to solve for x, we add 2 to both sides of the equation: x = 2 + ln(7) / ln(4). This expression gives us the exact solution for x. It's crucial to note that this solution is exact because it is expressed in terms of natural logarithms, which are irrational numbers. Using this form avoids any rounding errors that would occur if we converted the logarithms to decimal approximations too early. The exact solution is essential in many theoretical contexts and provides a benchmark for the accuracy of approximate solutions. In the next section, we will approximate this solution to four decimal places.

Approximate Solution to Four Decimal Places

To obtain an approximate solution to four decimal places for the equation 4^(x-2) = 7, we start with the exact solution we derived earlier: x = 2 + ln(7) / ln(4). The next step is to use a calculator to evaluate the natural logarithms and perform the division. The natural logarithm of 7, ln(7), is approximately 1.945910149, and the natural logarithm of 4, ln(4), is approximately 1.386294361. Therefore, the fraction ln(7) / ln(4) is approximately 1.945910149 / 1.386294361, which is about 1.403677466. Now, we add this value to 2 to find the approximate value of x: x ≈ 2 + 1.403677466 = 3.403677466. To round this result to four decimal places, we look at the fifth decimal place, which is 7. Since 7 is greater than or equal to 5, we round up the fourth decimal place. Thus, the approximate solution to four decimal places is x ≈ 3.4037. This approximation provides a practical value for x that can be used in applications where a decimal representation is necessary. It is important to remember that this is an approximation and differs slightly from the exact solution due to the rounding. However, for most practical purposes, an approximation to four decimal places is sufficiently accurate. This process demonstrates the utility of both exact and approximate solutions in mathematics.

Verification of the Solution

Verifying the solution is a critical step in solving any equation, including our exponential equation 4^(x-2) = 7. To verify our approximate solution, x ≈ 3.4037, we substitute this value back into the original equation and check if the equation holds true. We calculate 4^(3.4037-2) = 4^(1.4037). Using a calculator, we find that 4^(1.4037) is approximately 6.9998, which is very close to 7. The slight difference is due to the rounding we performed when approximating x to four decimal places. If we were to use the exact solution, x = 2 + ln(7) / ln(4), we would find that 4^((2 + ln(7) / ln(4)) - 2) is exactly 7, confirming the accuracy of the exact solution. This verification process highlights the importance of checking solutions, especially when approximations are involved. It ensures that our solution is valid and that any rounding errors have not significantly affected the result. Moreover, it reinforces our understanding of the equation and the solution process. Verification also helps in identifying any potential errors made during the solution process, such as incorrect application of logarithmic properties or mistakes in calculations. In summary, verifying the solution is an indispensable step in mathematical problem-solving.

Conclusion

In summary, we have successfully solved the exponential equation 4^(x-2) = 7 by finding both an exact solution and an approximate solution. We began by understanding the nature of exponential equations and the utility of logarithms in solving them. We then applied the natural logarithm to both sides of the equation, used the power rule of logarithms to simplify, and isolated the variable x. This process led us to the exact solution: x = 2 + ln(7) / ln(4). Next, we used a calculator to approximate the natural logarithms and found the decimal approximation to four decimal places: x ≈ 3.4037. Finally, we verified our solution by substituting the approximate value back into the original equation, confirming its accuracy. This exercise demonstrates the importance of both exact and approximate solutions in mathematics. Exact solutions provide a precise representation, while approximate solutions offer practical values for applications. Understanding the properties of logarithms and the methods for solving exponential equations is crucial in various fields, including mathematics, science, and engineering. This comprehensive approach ensures a thorough understanding of the problem and its solution. The combination of analytical methods and numerical approximations provides a powerful toolkit for tackling a wide range of mathematical problems.