Solving 4^(x-2) = 7 Exact And Approximate Solutions
Introduction
In this article, we will delve into solving the exponential equation 4^(x-2) = 7. This is a classic problem in mathematics that requires the use of logarithms to find both an exact solution and an approximate solution. Exponential equations are fundamental in various fields, including physics, engineering, and finance, making it crucial to understand how to solve them effectively. Our primary focus will be on employing the natural logarithm (ln) function to achieve this. We will first find the exact solution, which provides a symbolic representation of the answer. Then, we will use computational tools to find an approximate solution accurate to four decimal places, giving us a practical numerical answer. Understanding the process of solving exponential equations enhances problem-solving skills and provides a solid foundation for more advanced mathematical concepts.
Understanding Exponential Equations
Before diving into the specifics of our equation, let's briefly discuss exponential equations in general. An exponential equation is one in which the variable appears in the exponent. 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. Solving these equations often involves using logarithms, which are the inverse operations of exponentiation. Logarithms help us to "bring down" the exponent and isolate the variable. The common logarithm (log base 10) and the natural logarithm (log base e, denoted as ln) are frequently used in solving exponential equations. The choice of logarithm often depends on the specific problem and the need for numerical approximation. In our case, we will utilize the natural logarithm due to its widespread use in calculus and other advanced mathematical contexts. Understanding the properties of logarithms, such as the power rule, product rule, and quotient rule, is crucial for manipulating and solving exponential equations effectively.
Exact Solution Using the Natural Logarithm
To find the exact solution for the equation 4^(x-2) = 7, we will apply the natural logarithm to both sides of the equation. The natural logarithm, denoted as ln, is the logarithm to the base e, where e is an irrational number approximately equal to 2.71828. Applying the natural logarithm to both sides of the equation gives us: ln(4^(x-2)) = ln(7). Next, we use the power rule of logarithms, which states that ln(a^b) = b * ln(a). Applying this rule, we get: (x-2) * ln(4) = ln(7). Now, we need to isolate x. We start by dividing both sides of the equation by ln(4): x - 2 = ln(7) / ln(4). To solve for x, we add 2 to both sides: x = (ln(7) / ln(4)) + 2. This is the exact solution for x. It represents the value of x in terms of natural logarithms and constants, providing a precise algebraic expression. The exact solution is valuable because it avoids rounding errors that can occur when using decimal approximations. It also allows for further algebraic manipulation if needed. To summarize, the steps to find the exact solution involve applying the natural logarithm to both sides of the equation, using the power rule of logarithms to bring down the exponent, and then isolating the variable x.
Approximate Solution to Four Decimal Places
While the exact solution x = (ln(7) / ln(4)) + 2 is mathematically precise, it is often necessary to find an approximate numerical solution for practical applications. To find the approximate solution to four decimal places, we need to evaluate the expression using a calculator or computational tool. First, we calculate ln(7), which is approximately 1.945910149. Next, we calculate ln(4), which is approximately 1.386294361. Then, we divide ln(7) by ln(4): 1.945910149 / 1.386294361 ≈ 1.403677462. Finally, we add 2 to this result: 1.403677462 + 2 = 3.403677462. Rounding this to four decimal places, we get x ≈ 3.4037. This is the approximate solution of the equation 4^(x-2) = 7, accurate to four decimal places. The process of finding the approximate solution involves evaluating the natural logarithms, performing the necessary arithmetic operations, and rounding the result to the desired level of precision. Approximate solutions are particularly useful in real-world applications where numerical values are required.
Verification of the Solution
To ensure the accuracy of our solution, it is important to verify both the exact and approximate values. For the exact solution, we can substitute x = (ln(7) / ln(4)) + 2 back into the original equation 4^(x-2) = 7. Doing so, we have: 4^(((ln(7) / ln(4)) + 2) - 2) = 4^(ln(7) / ln(4)). Using the change of base formula for logarithms, we know that ln(7) / ln(4) = log_4(7). Therefore, the equation becomes 4^(log_4(7)), which simplifies to 7, confirming the exact solution. For the approximate solution, we substitute x ≈ 3.4037 into the original equation: 4^(3.4037 - 2) = 4^(1.4037). Evaluating this expression, we get approximately 7.0001, which is very close to 7, considering the rounding. This small discrepancy is due to the rounding of the decimal value. The verification process confirms that both the exact and approximate solutions are accurate. Verifying solutions is a critical step in problem-solving, as it helps to catch errors and build confidence in the results. It ensures that the mathematical manipulations performed are correct and that the solutions obtained are valid within the given context.
Applications of Exponential Equations
Exponential equations have numerous applications in various fields, including science, engineering, finance, and computer science. In finance, exponential functions are used to model compound interest and investment growth. The formula for compound interest is given by A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. In physics, exponential functions are used to describe radioactive decay, where the amount of a radioactive substance decreases exponentially over time. The equation for radioactive decay is N(t) = N_0 * e^(-λt), where N(t) is the amount of the substance at time t, N_0 is the initial amount, and λ is the decay constant. In biology, exponential functions are used to model population growth. The equation for exponential growth is P(t) = P_0 * e^(rt), where P(t) is the population at time t, P_0 is the initial population, and r is the growth rate. In computer science, exponential functions are used in the analysis of algorithms, particularly in the context of time complexity. For example, algorithms with exponential time complexity, such as brute-force algorithms, have running times that increase exponentially with the size of the input. These applications highlight the importance of understanding and solving exponential equations in various real-world scenarios.
Conclusion
In summary, we have successfully solved the exponential equation 4^(x-2) = 7 by finding both the exact solution and an approximate solution. The exact solution, x = (ln(7) / ln(4)) + 2, provides a precise algebraic representation of the answer, while the approximate solution, x ≈ 3.4037, offers a numerical value accurate to four decimal places. We utilized the natural logarithm function and the power rule of logarithms to solve the equation. Additionally, we verified both solutions to ensure their accuracy. Understanding the methods for solving exponential equations is crucial in mathematics and various applied fields, such as physics, engineering, finance, and computer science. Exponential equations are fundamental in modeling growth, decay, and other dynamic processes. The skills acquired in this process are valuable for tackling more complex mathematical problems and real-world applications. Mastering these concepts enhances problem-solving abilities and provides a solid foundation for further studies in mathematics and related disciplines. The ability to solve exponential equations is a key component of mathematical literacy, enabling individuals to analyze and interpret quantitative information effectively.