Solving 15e^x = 14: A Step-by-Step Guide

Jul 14, 2025 by ADMIN 41 views

Introduction

In this article, we will delve into the process of solving the exponential equation 15e^x = 14 for the variable x. This type of equation is commonly encountered in various fields, including mathematics, physics, engineering, and finance. Understanding how to solve exponential equations is crucial for modeling and analyzing real-world phenomena. We will explore the step-by-step method to isolate x, utilizing logarithms to obtain both the exact solution and an approximate solution rounded to four decimal places. Exponential equations play a significant role in modeling various phenomena, such as population growth, radioactive decay, compound interest, and the spread of infectious diseases. Mastering the techniques to solve these equations is essential for making accurate predictions and informed decisions in these fields. In this detailed exploration, we aim to provide a comprehensive understanding of the mathematical steps involved, as well as the underlying concepts that make this method effective. This will not only help in solving this specific equation but also in tackling similar problems in the future. The use of logarithms, particularly the natural logarithm, is a fundamental tool in solving exponential equations. By understanding the properties and applications of logarithms, one can effectively transform complex exponential problems into simpler algebraic forms. This article will guide you through each stage of the process, ensuring clarity and confidence in your ability to handle such equations. Let's embark on this mathematical journey to uncover the value of x that satisfies the given equation.

Step-by-Step Solution

To solve the equation 15e^x = 14, we will follow these steps:

Isolate the exponential term: Our first goal is to isolate the exponential term, which in this case is e^x. To do this, we need to divide both sides of the equation by 15:
```
15e^x / 15 = 14 / 15
e^x = 14/15
```
Isolating the exponential term is a critical first step because it sets the stage for using logarithms to solve for x. This process is analogous to isolating a variable in a linear equation before performing inverse operations. By dividing both sides by 15, we simplify the equation, making it easier to apply the logarithmic function. The resulting equation, e^x = 14/15, is now in a form where we can directly apply the natural logarithm to both sides, which is the next key step in finding the value of x. This step ensures that we are working with a clear and concise expression, which minimizes the chances of error in subsequent calculations. Furthermore, understanding this initial isolation is vital for tackling more complex exponential equations, where multiple terms might need to be rearranged before applying logarithms. The importance of this step cannot be overstated, as it forms the foundation for the rest of the solution.
Apply the natural logarithm: Now that we have isolated the exponential term, we can apply the natural logarithm (ln) to both sides of the equation. The natural logarithm is the inverse function of the exponential function with base e, which means that ln(e^x) = x. Applying the natural logarithm to both sides, we get:
```
ln(e^x) = ln(14/15)
x = ln(14/15)
```
The application of the natural logarithm is the pivotal step in solving exponential equations of this nature. The natural logarithm, denoted as ln, is the logarithm to the base e, where e is the mathematical constant approximately equal to 2.71828. The fundamental property of logarithms that makes this step effective is that ln(e^x) simplifies to x. This simplification allows us to directly solve for the exponent, which is otherwise trapped within the exponential function. By taking the natural logarithm of both sides of the equation, we transform the exponential equation into a linear equation in terms of x. This transformation is crucial because linear equations are much easier to solve. The result, x = ln(14/15), gives us the exact solution in terms of the natural logarithm. While this is a precise representation of the solution, it is often necessary to approximate the value using a calculator to get a numerical result. Understanding why and how the natural logarithm works is essential for solving a wide variety of exponential problems.
Exact value: The exact value of x is ln(14/15). This is the most precise representation of the solution, as it avoids any rounding errors. The exact value, x = ln(14/15), represents the most accurate solution to the equation. It's crucial in mathematical contexts where precision is paramount. The expression ln(14/15) is a symbolic representation that captures the true value of x without approximation. This form is particularly useful in theoretical work, where intermediate calculations need to be exact to ensure the final result is correct. For instance, if this value were to be used in further algebraic manipulations or in proving mathematical theorems, the exact form would be necessary to maintain integrity. Understanding the importance of exact values helps in appreciating the nuances of mathematical problem-solving. While approximate solutions are valuable for practical applications, the exact solution provides a deeper understanding of the underlying mathematical relationships. The ability to express and work with exact solutions is a hallmark of advanced mathematical proficiency. In the context of this problem, the exact value serves as a benchmark against which any approximate solutions can be compared, ensuring the accuracy of numerical methods.
Approximate value: To find the approximate value of x rounded to four decimal places, we can use a calculator to evaluate ln(14/15):
```
x ≈ -0.0690
```
The approximate value of x is obtained by evaluating the natural logarithm ln(14/15) using a calculator. This step is essential for practical applications where a numerical answer is required. Rounding the result to four decimal places, we get x ≈ -0.0690. This level of precision is often sufficient for engineering, scientific, and financial calculations. The process of finding the approximate value involves using a calculator that is capable of computing natural logarithms. Most scientific calculators have a dedicated ln function that makes this calculation straightforward. The act of rounding introduces a small amount of error, but it allows us to express the solution in a more manageable and interpretable form. In many real-world scenarios, the approximate value is more useful than the exact value because it provides a concrete number that can be easily understood and applied. For example, in a financial context, an interest rate calculated to four decimal places can be used to determine the return on an investment with a high degree of accuracy. Understanding the trade-offs between exact and approximate values is an important aspect of mathematical problem-solving.

Conclusion

In conclusion, the solution to the equation 15e^x = 14 is x = ln(14/15) exactly, or approximately x ≈ -0.0690 when rounded to four decimal places. Solving exponential equations often involves isolating the exponential term and applying logarithms, a fundamental technique in mathematics. The process of solving the exponential equation 15e^x = 14 highlights several key mathematical concepts. We began by isolating the exponential term, a crucial step that simplifies the equation and prepares it for the application of logarithms. Then, we applied the natural logarithm to both sides, leveraging its property as the inverse function of the exponential function with base e. This allowed us to transform the equation into a linear form, making it straightforward to solve for x. The exact solution, x = ln(14/15), provides a precise representation of the value, while the approximate solution, x ≈ -0.0690, offers a practical numerical answer. This exercise demonstrates the power and utility of logarithms in solving exponential equations, a technique that is widely applicable in various scientific and engineering disciplines. The ability to solve such equations is fundamental for modeling and analyzing a wide range of phenomena, from population growth and radioactive decay to financial investments and the spread of diseases. Mastering these techniques not only enhances one's mathematical skills but also provides a valuable tool for understanding and interpreting the world around us. The combination of analytical steps and numerical approximation provides a complete and practical approach to solving exponential equations. This skill is invaluable in both academic and professional settings, where mathematical modeling and problem-solving are essential.