Solve The Exponential Equation E^(2x-6) = 8^(x/2) And Express The Answer As Both An Exact Expression And A Decimal Approximation Rounded To Two Decimal Places.

When dealing with exponential equations, it's crucial to understand the properties of exponents and logarithms. This article delves into solving the exponential equation $e^{2x-6} = 8^{\frac{x}{2}}$, providing a step-by-step approach to finding both the exact expression and the decimal approximation rounded to two decimal places. We'll explore the underlying mathematical principles and techniques involved, ensuring a comprehensive understanding of the solution process. This exploration is crucial for anyone studying mathematics, particularly those delving into calculus, algebra, and other advanced mathematical topics where exponential equations frequently appear. Understanding how to solve these equations is not only academically beneficial but also has practical applications in fields such as finance, engineering, and computer science, where exponential models are used to describe various phenomena like growth, decay, and compound interest.

The primary challenge in solving exponential equations like this lies in manipulating the equation to isolate the variable x. To achieve this, we employ the properties of logarithms, which allow us to bring the exponent down as a coefficient. Logarithms are the inverse operations of exponentiation, making them powerful tools for solving equations where the variable is in the exponent. By applying logarithms to both sides of the equation, we can effectively linearize the problem, making it more amenable to algebraic manipulation. The choice of the base of the logarithm depends on the specific equation; in this case, the natural logarithm (base e) is particularly convenient due to the presence of e in the equation. However, it's also important to understand how to use other bases if needed, as some equations may require different approaches. The ultimate goal is to express x in terms of known constants, thereby providing both an exact solution and a decimal approximation for practical use. This process underscores the fundamental connection between exponential and logarithmic functions, which is a cornerstone of many mathematical and scientific applications.

1. Understanding the Equation

The given exponential equation is:

$$e^{2x-6} = 8^{\frac{x}{2}}$$

This equation involves exponential terms with different bases (e and 8), making it necessary to use logarithms to solve for x. Our goal is to express x in terms of elementary functions and then find its approximate value.

Understanding Exponential Equations

Before diving into the solution, it's essential to understand what exponential equations are and why they are significant. Exponential equations are mathematical expressions where the variable appears in the exponent. They are used extensively in various fields, including physics, engineering, finance, and computer science. For example, exponential functions can model population growth, radioactive decay, compound interest, and the charging/discharging of capacitors. The general form of an exponential equation is $a^{f(x)} = b^{g(x)}$, where a and b are constants and f(x) and g(x) are functions of x. Solving these equations often involves using logarithms, as logarithms are the inverse functions of exponentiation. This inverse relationship allows us to bring exponents down as coefficients, thereby simplifying the equation and making it solvable.

The significance of exponential equations lies in their ability to describe phenomena that change rapidly over time. For instance, in finance, the formula for compound interest involves an exponential term, reflecting the accelerating growth of an investment over time. Similarly, in physics, radioactive decay is modeled by an exponential equation, where the amount of radioactive material decreases exponentially with time. In biology, population growth under ideal conditions can be modeled by an exponential equation, where the population size increases rapidly. These examples highlight the broad applicability of exponential equations across various disciplines, making their study and understanding crucial for students and professionals alike. The techniques used to solve these equations, such as the application of logarithms, are fundamental tools in mathematical analysis and have far-reaching implications in modeling and understanding real-world phenomena. Therefore, mastering the methods for solving exponential equations is not just an academic exercise but a practical skill with significant real-world relevance.

2. Applying Natural Logarithms

To solve this exponential equation, we take the natural logarithm (ln) of both sides:

$$\ln(e^{2x-6}) = \ln(8^{\frac{x}{2}})$$

Using the property of logarithms that $\ln(a^b) = b \ln(a)$, we simplify the equation:

$$(2x-6)\ln(e) = \frac{x}{2}\ln(8)$$

Since $\ln(e) = 1$, the equation becomes:

$$2x - 6 = \frac{x}{2}\ln(8)$$

The Power of Natural Logarithms

The natural logarithm (ln) plays a pivotal role in solving exponential equations, particularly those involving the base e. The natural logarithm is the logarithm to the base e, where e is an irrational number approximately equal to 2.71828. Its importance stems from its unique properties and its frequent appearance in calculus and various scientific applications. One of the most crucial properties of the natural logarithm is its inverse relationship with the exponential function $e^x$. This relationship allows us to simplify equations where the variable is in the exponent, by bringing the exponent down as a coefficient.

When solving exponential equations, applying the natural logarithm to both sides is a common and effective strategy. This is because the logarithmic function is the inverse of the exponential function, which simplifies the equation by "undoing" the exponentiation. For instance, if we have an equation of the form $e^{f(x)} = g(x)$, taking the natural logarithm of both sides gives us $\ln(e^{f(x)}) = \ln(g(x))$. Using the property that $\ln(e^u) = u$, the equation simplifies to $f(x) = \ln(g(x))$, which is often easier to solve. This transformation is particularly useful when the exponent is a function of x, as it allows us to isolate and solve for x. Moreover, the natural logarithm is not only useful for equations involving e but can also be applied to any exponential equation by using the change of base formula, if necessary. Understanding the properties and applications of the natural logarithm is therefore essential for solving a wide range of mathematical problems, especially in calculus, differential equations, and various fields of applied science and engineering.

3. Solving for x

Now we solve the linear equation for x:

$$2x - 6 = \frac{x}{2}\ln(8)$$

Multiply both sides by 2 to eliminate the fraction:

$$4x - 12 = x\ln(8)$$

Rearrange the equation to isolate x terms:

$$4x - x\ln(8) = 12$$

Factor out x:

$$x(4 - \ln(8)) = 12$$

Divide by $(4 - \ln(8))$ to solve for x:

$$x = \frac{12}{4 - \ln(8)}$$

Isolating the Variable: Techniques and Strategies

Isolating the variable is a fundamental step in solving almost any type of equation, and exponential equations are no exception. The process involves manipulating the equation using algebraic techniques to get the variable alone on one side. In the context of exponential equations, this often means dealing with exponents and logarithms. The techniques used to isolate the variable can vary depending on the specific equation, but there are several common strategies that are frequently employed.

One primary strategy is to use inverse operations. Since exponential and logarithmic functions are inverses of each other, applying logarithms to an exponential equation or exponentiating a logarithmic equation can help simplify the expression. For example, if we have an equation of the form $a^{f(x)} = b$, we can take the logarithm of both sides to get $f(x) \log(a) = \log(b)$, which allows us to isolate the function f(x). Similarly, if we have an equation of the form $\log_a(f(x)) = c$, we can exponentiate both sides to get $f(x) = a^c$. Another common technique is to use algebraic manipulation to rearrange the equation. This might involve adding or subtracting terms, multiplying or dividing both sides by a constant, or factoring out common factors. The goal is to group terms involving the variable on one side of the equation and constant terms on the other side. This process often requires a combination of steps, and the most effective approach can depend on the structure of the equation. Practicing these techniques and understanding the underlying principles is essential for becoming proficient in solving a wide variety of mathematical problems.

4. Exact Expression

The exact expression for x is:

$$x = \frac{12}{4 - \ln(8)}$$

This is the exact solution in terms of elementary functions. An exact expression is crucial because it represents the solution without any approximation errors. It allows for precise calculations and further manipulation if needed. This exact form is particularly useful in theoretical contexts where the symbolic representation is more important than a numerical value. Moreover, it can serve as a benchmark to verify the accuracy of numerical approximations. In the realm of mathematics and its applications, the exact expression often provides deeper insights into the nature of the solution and its relationship to the parameters of the problem.

The Significance of Exact Expressions in Mathematics

In mathematics, an exact expression is a solution presented in terms of known constants and elementary functions, without resorting to approximations. The significance of obtaining an exact expression lies in its precision and the ability to represent the solution in a symbolic form. Unlike decimal approximations, which inherently involve rounding errors, an exact expression retains the full mathematical relationship between the variables and parameters of the problem. This precision is crucial in many theoretical and practical contexts, where even small errors can propagate and lead to significant discrepancies in the final result. For instance, in scientific computations, engineering designs, and financial modeling, the use of exact expressions can ensure the accuracy and reliability of calculations.

Furthermore, exact expressions often provide deeper insights into the underlying mathematical structure of a problem. They can reveal patterns, symmetries, and relationships that might not be apparent from numerical approximations alone. For example, an exact solution might involve irrational numbers or trigonometric functions, which can highlight the nature of the solution and its connection to other mathematical concepts. Additionally, exact expressions are essential for symbolic manipulation and further analysis. They can be used to derive other results, prove theorems, and perform algebraic operations without the limitations imposed by numerical approximations. This is particularly important in fields such as theoretical physics, where equations are often manipulated symbolically to gain a deeper understanding of physical phenomena. Therefore, while numerical approximations have their place, the pursuit of exact expressions remains a central goal in mathematics and its applications, providing a foundation for precision, understanding, and further exploration.

5. Decimal Approximation

To find the decimal approximation, we use the given value $e = 2.71828182845905$ and calculate $\ln(8)$. Since $8 = 2^3$, $\ln(8) = 3\ln(2)$. We know that $\ln(e) = 1$, so we can approximate $\ln(2)$ using the fact that $e^{0.693} \approx 2$. Therefore, $\ln(2) \approx 0.693$, and $\ln(8) \approx 3 \times 0.693 = 2.079$.

Now we substitute this value into the expression for x:

$$x = \frac{12}{4 - 2.079} = \frac{12}{1.921} \approx 6.24778761062988$$

Rounded to two decimal places, the decimal approximation is:

$$x \approx 6.25$$

Approximations and Their Importance in Practical Applications

While exact expressions are crucial for theoretical accuracy, decimal approximations play a vital role in practical applications. In many real-world scenarios, a numerical answer is required, and the exact expression, while mathematically precise, may not be directly usable. For instance, in engineering, physics, and finance, solutions often need to be implemented or interpreted in a numerical form. Decimal approximations provide a way to bridge the gap between theoretical solutions and practical applications, allowing us to quantify the results and make informed decisions.

Approximations are also essential when dealing with irrational or transcendental numbers, such as π or e. These numbers cannot be expressed as a finite decimal or fraction, so we must rely on approximations to work with them. The level of approximation needed depends on the context. In some cases, a rough estimate may suffice, while in others, high precision is necessary. For example, in calculations involving the global positioning system (GPS), even small errors in the numerical values can lead to significant inaccuracies in location. Therefore, it is crucial to understand the trade-offs between accuracy and computational efficiency when choosing an appropriate level of approximation.

Moreover, the process of finding decimal approximations often involves numerical methods and algorithms, which are valuable tools in their own right. These methods allow us to solve problems that may not have an exact solution or for which the exact solution is too complex to compute directly. Numerical analysis is a field of mathematics that focuses on the development and analysis of these methods, and it plays a crucial role in many scientific and engineering disciplines. Thus, while exact solutions are ideal, approximations are often necessary and valuable in translating mathematical results into practical actions.

6. Final Answer

The solution to the exponential equation $e^{2x-6} = 8^{\frac{x}{2}}$ is:

• Exact Expression: $x = \frac{12}{4 - \ln(8)}$
• Decimal Approximation: $x \approx 6.25$

This detailed solution demonstrates the process of solving exponential equations using logarithms and provides both the exact and approximate solutions, which are essential in various mathematical and practical contexts.

Recapitulation of Solving Exponential Equations

In summary, solving exponential equations involves a series of strategic steps, each designed to simplify the equation and isolate the variable. The process typically begins with recognizing the structure of the equation and identifying the presence of exponential terms. The next crucial step is to apply logarithms to both sides of the equation. This is often done using the natural logarithm (ln) due to its convenient properties, particularly when dealing with the base e. However, the choice of the logarithm base can depend on the specific equation, and other bases can be used when appropriate.

Once logarithms are applied, the properties of logarithms are used to simplify the equation. This usually involves bringing exponents down as coefficients, which effectively linearizes the equation. At this stage, the equation is transformed into a form that can be solved using algebraic techniques. These techniques may include rearranging terms, factoring, and using inverse operations to isolate the variable. The goal is to express the variable in terms of known constants and functions. The result is an exact expression for the solution, which is a precise representation of the solution in terms of elementary functions.

In many practical applications, a decimal approximation is also needed. This involves evaluating the exact expression numerically, which may require using a calculator or computer. Approximations are particularly useful when dealing with irrational or transcendental numbers or when the solution needs to be implemented in a real-world context. However, it is important to be mindful of the level of approximation and the potential for rounding errors. The final solution is presented in both exact expression and decimal approximation forms, providing a complete and practical answer to the original exponential equation. This comprehensive approach underscores the interplay between theoretical precision and practical usability in mathematical problem-solving.