Solving The Exponential Equation 11e^x - 14 = 3 A Step-by-Step Guide

Introduction

In the realm of mathematics, exponential equations hold a significant place, often appearing in various scientific and engineering applications. These equations involve variables in the exponents, making them distinct from polynomial equations. Solving them requires a different set of techniques and understanding of the properties of exponential functions. In this comprehensive guide, we will delve into the process of finding solutions for a specific exponential equation: 11e^x - 14 = 3. We will explore the step-by-step approach to isolate the variable x, express the exact solution in terms of the natural logarithm, and then approximate the solution to four decimal places. This exploration will not only enhance your understanding of exponential equations but also equip you with the skills to tackle similar problems.

Understanding Exponential Equations

Before we dive into solving the equation, it's crucial to grasp the fundamental concepts behind exponential equations. An exponential equation is an equation in which the variable appears in the exponent. The general form of an exponential equation can be expressed as a^x = b, where 'a' is the base, 'x' is the exponent, 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 doesn't lead to interesting exponential relationships. The exponent 'x' can be any real number. The key to solving exponential equations lies in isolating the exponential term and then using logarithms to bring the exponent down. Logarithms are the inverse functions of exponentials, meaning they "undo" the exponential operation. The most commonly used logarithms are the natural logarithm (ln), which has a base of 'e' (Euler's number, approximately 2.71828), and the common logarithm (log), which has a base of 10. When dealing with exponential equations involving the base 'e', the natural logarithm is particularly useful.

Step-by-Step Solution of 11e^x - 14 = 3

Let's now embark on a step-by-step journey to solve the exponential equation 11e^x - 14 = 3. Our primary goal is to isolate the term containing the exponent, which in this case is 11e^x. This involves a series of algebraic manipulations to gradually simplify the equation.

Step 1: Isolate the Exponential Term

To begin, we need to isolate the exponential term, 11e^x. This can be achieved by adding 14 to both sides of the equation. This operation maintains the balance of the equation while moving the constant term to the right side:

11e^x - 14 + 14 = 3 + 14

This simplifies to:

11e^x = 17

Step 2: Divide to Isolate e^x

Next, we need to isolate e^x by dividing both sides of the equation by 11. This will leave the exponential term alone on the left side:

(11e^x) / 11 = 17 / 11

This simplifies to:

e^x = 17/11

Step 3: Apply the Natural Logarithm

Now that we have isolated e^x, we can use the natural logarithm to solve for x. The natural logarithm, denoted as ln, is the inverse function of the exponential function with base e. Applying the natural logarithm to both sides of the equation will "undo" the exponential:

ln(e^x) = ln(17/11)

Using the property of logarithms that ln(a^b) = b * ln(a), we get:

x * ln(e) = ln(17/11)

Since ln(e) = 1, the equation simplifies to:

x = ln(17/11)

Step 4: Exact Solution

The exact solution for x is therefore ln(17/11). This is an exact representation of the solution, expressed in terms of the natural logarithm. It's important to note that this is the most precise way to represent the solution, as it avoids any rounding errors that might occur when approximating the value.

Step 5: Approximate Solution

To obtain an approximate solution accurate to four decimal places, we can use a calculator to evaluate ln(17/11). Using a calculator, we find that:

x ≈ 0.4331

Therefore, the approximate solution to the equation 11e^x - 14 = 3, accurate to four decimal places, is 0.4331.

Expressing the Exact Solution in Terms of the Natural Logarithm

As we found in the previous section, the exact solution to the exponential equation 11e^x - 14 = 3 is x = ln(17/11). This expression utilizes the natural logarithm, which is the logarithm to the base e, where e is Euler's number (approximately 2.71828). The natural logarithm is the inverse function of the exponential function with base e, making it an indispensable tool for solving exponential equations where e is the base.

To fully appreciate the significance of expressing the solution in terms of the natural logarithm, let's delve deeper into the properties and implications of this representation. The natural logarithm provides a precise and concise way to represent the solution without introducing any rounding errors. It encapsulates the exact relationship between the exponential term and the variable x. In many mathematical and scientific contexts, the exact solution is preferred over an approximate solution, as it preserves the integrity of the mathematical relationship and allows for further analysis and manipulation without the accumulation of errors.

The expression ln(17/11) can also be further manipulated using the properties of logarithms. One such property is the quotient rule, which states that ln(a/b) = ln(a) - ln(b). Applying this rule to our solution, we can rewrite it as:

x = ln(17) - ln(11)

This alternative representation of the exact solution highlights the difference between the natural logarithms of 17 and 11. While this form is mathematically equivalent to ln(17/11), it might be more convenient in certain situations, such as when you need to work with the individual logarithms separately.

In conclusion, expressing the exact solution in terms of the natural logarithm not only provides a precise answer but also allows for further manipulation and analysis using the properties of logarithms. It underscores the importance of understanding and utilizing the natural logarithm when dealing with exponential equations.

Approximating the Solution to Four Decimal Places

While the exact solution, expressed as x = ln(17/11), provides a mathematically precise representation, in practical applications, we often require a numerical approximation. This is where approximating the solution to a certain number of decimal places becomes crucial. In this case, we aim to approximate the solution to four decimal places. To achieve this, we need to evaluate the natural logarithm using a calculator or a computer software capable of handling logarithmic functions.

Using a calculator, we input ln(17/11) and obtain a decimal value. The calculator typically provides a result with several decimal places. To approximate the solution to four decimal places, we focus on the first five decimal places. If the fifth decimal place is 5 or greater, we round up the fourth decimal place. If the fifth decimal place is less than 5, we leave the fourth decimal place as it is.

When we evaluate ln(17/11) using a calculator, we get a result approximately equal to 0.433074636. Now, let's apply the rounding rule to approximate this value to four decimal places. The fifth decimal place is 7, which is greater than 5. Therefore, we round up the fourth decimal place (0) to 1. This gives us the approximate solution:

x ≈ 0.4331

This approximate solution is accurate to four decimal places, meaning that the error between this approximation and the exact solution is less than 0.00005. In many real-world scenarios, this level of accuracy is sufficient. It's important to remember that an approximate solution is not the exact solution, but it provides a practical and usable value for most applications.

In summary, approximating the solution to four decimal places involves evaluating the natural logarithm using a calculator and applying the appropriate rounding rule. The resulting value, x ≈ 0.4331, provides a practical approximation of the solution to the exponential equation 11e^x - 14 = 3.

Applications of Exponential Equations

Exponential equations are not just theoretical constructs; they have a wide range of practical applications in various fields. Understanding how to solve them is crucial for modeling and analyzing real-world phenomena. Let's explore some key areas where exponential equations play a significant role.

1. Population Growth and Decay

One of the most common applications of exponential equations is in modeling population dynamics. The growth of a population (whether it's humans, animals, or bacteria) can often be described by an exponential function. Similarly, the decay of a population due to factors like disease or resource scarcity can also be modeled using exponential equations. The general formula for exponential growth is:

N(t) = N₀ * e^(rt)

Where:

• N(t) is the population at time t
• N₀ is the initial population
• r is the growth rate (positive for growth, negative for decay)
• e is the base of the natural logarithm (approximately 2.71828)
• t is the time

This equation can be used to predict future population sizes, estimate growth rates, or determine the time it takes for a population to reach a certain size. For example, if we know the initial population of a bacteria colony and its growth rate, we can use this equation to predict how many bacteria there will be after a certain amount of time.

2. Radioactive Decay

Radioactive decay is another phenomenon that is accurately described by exponential equations. Radioactive substances decay over time, meaning that the amount of the substance decreases exponentially. The rate of decay is characterized by the half-life, which is the time it takes for half of the substance to decay. The formula for radioactive decay is:

N(t) = N₀ * e^(-λt)

Where:

• N(t) is the amount of the substance remaining at time t
• N₀ is the initial amount of the substance
• λ is the decay constant (related to the half-life)
• e is the base of the natural logarithm
• t is the time

This equation is used in various applications, including carbon dating (determining the age of ancient artifacts), medical imaging (using radioactive isotopes as tracers), and nuclear power generation.

3. Compound Interest

In the realm of finance, exponential equations are fundamental to understanding compound interest. Compound interest is the interest earned on both the principal amount and the accumulated interest. The formula for compound interest is:

A = P(1 + r/n)^(nt)

Where:

• A is the final amount after t years
• P is the principal amount (the initial investment)
• r is the annual interest rate (as a decimal)
• n is the number of times the interest is compounded per year
• t is the number of years

This equation shows that the amount of money grows exponentially over time, especially when the interest is compounded frequently. Understanding compound interest is crucial for making informed financial decisions, such as investing, saving for retirement, and taking out loans.

4. Learning Curves

Exponential equations can also be used to model learning curves, which describe how quickly someone learns a new skill or concept. In many cases, the rate of learning decreases over time, meaning that the initial progress is rapid, but it slows down as the person becomes more proficient. A typical learning curve can be represented by the equation:

L(t) = L∞ - (L∞ - L₀) * e^(-kt)

Where:

• L(t) is the level of learning at time t
• L∞ is the maximum level of learning that can be achieved
• L₀ is the initial level of learning
• k is the learning rate constant
• e is the base of the natural logarithm
• t is the time

This equation can be used to predict how long it will take someone to reach a certain level of proficiency, or to compare the effectiveness of different learning methods.

Conclusion

In this comprehensive guide, we have explored the solution of the exponential equation 11e^x - 14 = 3. We meticulously walked through the steps to isolate the variable x, express the exact solution in terms of the natural logarithm (x = ln(17/11)), and approximate the solution to four decimal places (x ≈ 0.4331). We also emphasized the importance of understanding the properties of logarithms and their role in solving exponential equations. Furthermore, we discussed the wide-ranging applications of exponential equations in various fields, including population growth, radioactive decay, compound interest, and learning curves.

Mastering the techniques to solve exponential equations is a valuable skill in mathematics and its applications. The ability to manipulate exponential expressions, apply logarithms, and interpret the solutions in real-world contexts is crucial for problem-solving and decision-making in various scientific, engineering, and financial scenarios. This guide serves as a foundation for further exploration of exponential functions and their applications, empowering you to tackle more complex problems and gain a deeper understanding of the world around us. By understanding the underlying principles and practicing the techniques, you can confidently approach exponential equations and unlock their power to model and analyze a wide array of phenomena.