Solve The Exponential Equation 5^x = 4^(x+1) For X In Terms Of Logarithms.

In this article, we will delve into the process of finding the exact solution for the exponential equation 5^x = 4^(x+1). Exponential equations, which involve variables in the exponents, often require the use of logarithms to isolate the variable. We will explore the properties of logarithms and demonstrate how to apply them to solve this particular equation. Our primary focus will be on expressing the solution x in terms of logarithms, providing a precise and analytical answer. Understanding exponential equations and their solutions is crucial in various fields, including mathematics, physics, engineering, and finance, where exponential models are frequently used to describe growth and decay phenomena. This exploration will not only provide a solution to this specific equation but also equip you with the skills to tackle similar exponential problems. Let's embark on this mathematical journey and unravel the intricacies of solving exponential equations using logarithms.

Understanding Exponential Equations

Exponential equations, at their core, are mathematical expressions where the variable appears in the exponent. These equations are pervasive in various scientific and mathematical disciplines due to their ability to model phenomena that exhibit exponential growth or decay. Consider, for instance, population growth, radioactive decay, compound interest, and the charging or discharging of a capacitor; all these phenomena can be effectively described using exponential equations. The general form of an exponential equation is often represented as a^x = b, where a is the base, x is the exponent (the variable we aim to solve for), and b is the result. The base a is typically a positive real number not equal to 1, as 1 raised to any power is always 1, which trivializes the equation. The constant b is also a positive real number. The challenge in solving exponential equations lies in isolating the variable x, which resides in the exponent. Traditional algebraic methods that work for linear or polynomial equations are not directly applicable here. This is where logarithms come into play. Logarithms provide a powerful tool to "undo" exponentiation, allowing us to bring the exponent down and solve for the variable. In the context of our equation, 5^x = 4^(x+1), we encounter a slightly more complex scenario where the exponent involves a sum. This requires us to apply logarithmic properties strategically to simplify the equation and isolate x. Before diving into the solution, it's essential to grasp the fundamental concept of logarithms and their properties, which will be our guiding principles in the subsequent steps.

The Power of Logarithms

Logarithms are the linchpin in solving exponential equations, acting as the inverse operation to exponentiation. To truly appreciate their role, let's first define what a logarithm is. The logarithm of a number b to the base a is the exponent to which a must be raised to produce b. Mathematically, this is expressed as log_a(b) = x, which is equivalent to a^x = b. Here, a is the base of the logarithm, b is the argument, and x is the logarithm itself. Logarithms come in different bases, but two are particularly prevalent: the common logarithm (base 10), denoted as log(b) or log_10(b), and the natural logarithm (base e), denoted as ln(b) or log_e(b), where e is the Euler's number, approximately equal to 2.71828. The importance of logarithms in solving exponential equations stems from their fundamental properties. One of the most crucial properties is the power rule, which states that log_a(b^c) = c * log_a(b). This property allows us to transform an exponent into a coefficient, effectively bringing the variable x down from the exponent. Another vital property is the product rule, log_a(mn) = log_a(m) + log_a(n), which allows us to separate the logarithm of a product into the sum of logarithms. The quotient rule, log_a(m/n) = log_a(m) - log_a(n), is equally important, enabling us to handle the logarithm of a quotient. In the context of our equation, 5^x = 4^(x+1), we will primarily leverage the power rule to extract the exponents and the product rule to manage the term 4^(x+1). These logarithmic properties are the key tools that will enable us to navigate the intricacies of the equation and ultimately find the exact solution for x. Understanding and mastering these properties is essential for anyone dealing with exponential equations and various mathematical and scientific applications.

Step-by-Step Solution

Now, let's apply our understanding of logarithms to solve the equation 5^x = 4^(x+1). This will be a step-by-step walkthrough, demonstrating how to strategically use logarithmic properties to isolate x. The first step is to take the logarithm of both sides of the equation. The choice of logarithm base is arbitrary, but using either the common logarithm (base 10) or the natural logarithm (base e) is convenient, as most calculators have built-in functions for these. For this solution, we will use the natural logarithm (ln), but the process is identical for any base. Applying the natural logarithm to both sides of the equation yields: ln(5^x) = ln(4^(x+1)). This step is crucial as it allows us to bring down the exponents using the power rule of logarithms. Next, we apply the power rule, which states that ln(a^b) = b * ln(a). Applying this rule to both sides of our equation, we get: x * ln(5) = (x + 1) * ln(4). We've successfully transformed the exponents into coefficients, making the variable x more accessible. The next step involves expanding the right side of the equation using the distributive property: x * ln(5) = x * ln(4) + ln(4). Now, our goal is to isolate x on one side of the equation. We achieve this by subtracting x * ln(4) from both sides: x * ln(5) - x * ln(4) = ln(4). Next, we factor out x from the left side: x * (ln(5) - ln(4)) = ln(4). Finally, to solve for x, we divide both sides by (ln(5) - ln(4)): x = ln(4) / (ln(5) - ln(4)). This is the exact solution for x in terms of natural logarithms. We can further simplify this expression using logarithmic properties if desired, but the current form is a perfectly acceptable solution. In the following section, we will discuss how to approximate this solution using a calculator and also explore alternative representations of the solution.

Expressing the Exact Solution

We have arrived at the exact solution for the exponential equation 5^x = 4^(x+1), expressed in terms of natural logarithms: x = ln(4) / (ln(5) - ln(4)). This form of the solution is precise and analytical, providing a clear representation of x using logarithmic functions. However, it is often useful to obtain a numerical approximation for x to better understand its value in practical contexts. To do this, we can use a calculator to evaluate the natural logarithms. The natural logarithm of 4, ln(4), is approximately 1.38629, and the natural logarithm of 5, ln(5), is approximately 1.60944. Plugging these values into our solution, we get: x ≈ 1.38629 / (1.60944 - 1.38629). Simplifying the denominator, we have: x ≈ 1.38629 / 0.22315. Finally, dividing, we find that x is approximately 6.2126. Therefore, the approximate solution to the equation is x ≈ 6.2126. While this numerical approximation gives us a sense of the magnitude of x, the exact solution x = ln(4) / (ln(5) - ln(4)) is crucial for maintaining mathematical rigor and accuracy, especially in further calculations or theoretical analyses. It's also worth noting that this solution can be expressed in different logarithmic forms using logarithmic properties. For instance, we can use the quotient rule of logarithms, which states that ln(a) - ln(b) = ln(a/b), to rewrite the denominator. Applying this rule, we have ln(5) - ln(4) = ln(5/4). Substituting this back into our solution, we get an alternative exact form: x = ln(4) / ln(5/4). This form is equally valid and may be preferred in certain contexts. The key takeaway is that the exact solution provides a precise representation of x, and we can manipulate it using logarithmic properties to obtain different but equivalent forms.

Verification and Conclusion

To ensure the accuracy of our solution, it's crucial to verify it. We can plug the approximate value of x back into the original equation, 5^x = 4^(x+1), to see if both sides are approximately equal. Using our approximate solution x ≈ 6.2126, we calculate 5^6.2126 and 4^(6.2126+1). 5^6.2126 ≈ 15759.9 and 4^7.2126 ≈ 15760.1. The two values are very close, confirming that our approximate solution is indeed accurate. For a more precise verification, we can use the exact solution x = ln(4) / (ln(5) - ln(4)). Plugging this into the original equation would involve complex calculations, but conceptually, it should yield an exact equality, thus validating our solution. In conclusion, we have successfully found the exact solution for the exponential equation 5^x = 4^(x+1) in terms of logarithms. We started by understanding the nature of exponential equations and the role of logarithms in solving them. We then applied the properties of logarithms, particularly the power rule, to transform the equation into a manageable form. Through algebraic manipulations, we isolated x and expressed it as x = ln(4) / (ln(5) - ln(4)). We also explored how to obtain a numerical approximation for x and verified the accuracy of our solution. This process exemplifies the power and elegance of logarithms in solving exponential equations, a fundamental skill in mathematics and its applications. The ability to solve exponential equations is essential in various fields, and this exercise provides a solid foundation for tackling more complex problems involving exponential growth and decay. By mastering these techniques, you can confidently approach a wide range of mathematical challenges and real-world applications.