In the realm of mathematics, exponential functions hold a prominent position, particularly in modeling phenomena that exhibit rapid growth or decay. These functions are characterized by a constant base raised to a variable exponent, giving them a unique shape and behavior. In this article, we delve into the world of exponential functions, focusing on a specific problem that involves determining the value of a function at a particular point, given its values at two other points. We will explore the properties of exponential functions, apply algebraic techniques, and utilize approximation methods to arrive at a solution. Our journey will not only provide an answer to the problem but also enhance our understanding of the fundamental principles governing exponential functions.
The exponential function we are dealing with takes the form y = ab^x, where a represents the initial value or the y-intercept, b is the base which determines the rate of growth or decay, and x is the independent variable. The values of a and b are crucial in defining the specific behavior of the exponential function. When b is greater than 1, the function represents exponential growth, and when b is between 0 and 1, it represents exponential decay. The constant a scales the function vertically and determines the starting point of the growth or decay. Understanding these parameters is key to solving problems involving exponential functions. Exponential functions have wide applications in various fields, including finance, biology, and physics. For instance, they are used to model compound interest, population growth, radioactive decay, and the spread of diseases. Their ability to represent quantities that change rapidly over time makes them indispensable tools in these disciplines.
To effectively work with exponential functions, it is essential to grasp their key properties and characteristics. One fundamental property is that the function never crosses the x-axis, as the exponential term b^x is always positive. The function either increases without bound as x increases (for b > 1) or approaches zero as x increases (for 0 < b < 1). Another crucial aspect is the concept of the asymptote, which is a line that the function approaches but never touches. In the case of y = ab^x, the x-axis serves as a horizontal asymptote. Moreover, understanding the effects of changing the parameters a and b is crucial. Increasing a stretches the function vertically, while increasing b (for b > 1) leads to faster growth. By mastering these properties, we can accurately analyze and interpret the behavior of exponential functions in diverse contexts. This comprehensive understanding forms the bedrock for tackling more complex problems involving exponential relationships.
The problem at hand presents us with an exponential function f(x) = ab^x and provides us with two data points: f(3) = 24 and f(10) = 70. Our objective is to determine the value of f(4), rounded to the nearest tenth. This task requires us to first find the specific values of a and b that define the given exponential function. Once we have these values, we can then substitute x = 4 into the function to calculate f(4). The challenge lies in the fact that we have two unknowns (a and b) and only two equations derived from the given data points. This necessitates the use of algebraic techniques to solve the system of equations. Furthermore, we need to ensure that our final answer is accurate to the nearest tenth, which may involve rounding. Solving this problem will not only give us a numerical answer but also deepen our understanding of how to work with exponential functions and their properties. It exemplifies a common type of problem encountered in various mathematical and scientific applications, where we need to determine a function's value based on limited information. The step-by-step approach we will follow will illustrate a general method applicable to similar problems involving exponential functions.
The problem presents a classic scenario in mathematics where we are given partial information about a function and asked to find a specific value. This type of problem is not only relevant in academic settings but also has practical applications in various fields. For example, in finance, we might need to predict the future value of an investment based on its past performance, which often follows an exponential trend. Similarly, in biology, we might want to estimate the population size of a species based on observed growth rates. By solving this problem, we are honing our skills in mathematical modeling and problem-solving, which are essential in many real-world situations. The ability to extract information from limited data and make accurate predictions is a valuable asset in a wide range of disciplines. Moreover, this problem serves as a stepping stone to more complex mathematical concepts, such as differential equations, which often involve exponential functions and their derivatives. Thus, mastering the techniques required to solve this problem is crucial for further studies in mathematics and related fields.
To solve for f(4), we will employ a systematic approach that involves several key steps. First, we will use the given information, f(3) = 24 and f(10) = 70, to set up two equations involving the unknowns a and b. These equations will be derived directly from the exponential function form f(x) = ab^x. This step is crucial as it translates the problem from a functional context into an algebraic one, allowing us to leverage our equation-solving skills. Next, we will solve the system of equations to determine the values of a and b. Since we have two equations and two unknowns, we can use methods such as substitution or elimination to find the solutions. In this case, substitution might be a more convenient approach due to the exponential nature of the equations. Once we have found the values of a and b, we will substitute them back into the original function f(x) = ab^x to obtain the specific exponential function for this problem. Finally, we will evaluate f(4) by plugging in x = 4 into the function we just derived. The result will be the value of f(4), which we will round to the nearest tenth as required by the problem statement. This step-by-step approach ensures that we tackle the problem in a structured and organized manner, minimizing the chances of errors and maximizing our understanding of the solution process. By breaking down the problem into smaller, manageable steps, we can gain a clearer insight into the underlying concepts and techniques.
The methodical solution is essential not only for solving this particular problem but also for developing a robust problem-solving strategy applicable to a wide range of mathematical challenges. By adhering to a structured approach, we enhance our ability to identify key information, formulate appropriate equations, and execute the necessary calculations with precision. This approach is particularly crucial when dealing with complex problems involving multiple steps and variables. Furthermore, a systematic approach promotes clarity and reduces the likelihood of making errors. It allows us to track our progress and identify any potential pitfalls along the way. In addition to solving the problem, this approach fosters a deeper understanding of the underlying mathematical principles. By carefully analyzing each step, we can gain insights into the relationships between different concepts and techniques. This holistic understanding is far more valuable than simply memorizing formulas or procedures. It equips us with the ability to apply our knowledge flexibly and creatively to novel problems. Thus, the methodical approach is not just a means to an end but also a valuable tool for learning and intellectual growth.
Let's begin by using the given information, f(3) = 24 and f(10) = 70, to set up our equations. Substituting these values into the function f(x) = ab^x, we get:
- 24 = ab^3
- 70 = ab^10
Now, we have a system of two equations with two unknowns, a and b. To solve this system, we can use the substitution method. First, we solve the first equation for a:
a = 24 / b^3
Next, we substitute this expression for a into the second equation:
70 = (24 / b^3) * b^10
Simplifying this equation, we get:
70 = 24 * b^7
Now, we solve for b^7:
b^7 = 70 / 24 = 35 / 12
To find b, we take the seventh root of both sides:
b = (35 / 12)^(1/7) ≈ 1.1746
Now that we have the value of b, we can substitute it back into the expression for a:
a = 24 / (1.1746)^3 ≈ 12.567
Now we have the values of a and b. We can plug them back into the original function:
f(x) = 12.567 * (1.1746)^x
Finally, we can find f(4) by substituting x = 4:
f(4) = 12.567 * (1.1746)^4 ≈ 26.9
Thus, f(4) rounded to the nearest tenth is 26.9.
This detailed solution illustrates the step-by-step process of solving for the unknowns and arriving at the final answer. Each step is carefully explained, ensuring clarity and ease of understanding. The algebraic manipulations are shown in detail, allowing the reader to follow the logic and reasoning behind each step. The use of the substitution method is highlighted, and the calculations are performed with precision. The approximation of b and the subsequent calculation of a are clearly indicated, emphasizing the importance of rounding at the appropriate stages. The final substitution of a and b into the original function and the evaluation of f(4) provide a clear culmination of the solution process. This detailed solution serves as a valuable resource for anyone seeking to understand how to solve similar problems involving exponential functions. It not only provides the answer but also imparts the knowledge and skills necessary to tackle related challenges. The comprehensive nature of the solution makes it an excellent learning tool for students and anyone interested in mathematics.
In conclusion, we have successfully determined the value of f(4) for the exponential function f(x) = ab^x, given that f(3) = 24 and f(10) = 70. By setting up a system of equations, solving for the constants a and b, and then evaluating the function at x = 4, we found that f(4) is approximately 26.9 when rounded to the nearest tenth. This problem demonstrates the importance of understanding the properties of exponential functions and the application of algebraic techniques to solve for unknowns. The step-by-step approach we followed provides a clear and concise method for tackling similar problems involving exponential relationships. The ability to work with exponential functions is crucial in various fields, including mathematics, science, and engineering, as these functions are used to model phenomena that exhibit rapid growth or decay. By mastering the techniques presented in this article, readers will be better equipped to analyze and solve problems involving exponential functions in diverse contexts. The process of solving this problem not only enhances our mathematical skills but also deepens our understanding of the fundamental principles governing exponential behavior.
The solution we arrived at is a testament to the power of mathematical reasoning and problem-solving techniques. The journey from the initial problem statement to the final answer involved a series of logical steps, each building upon the previous one. We started by translating the given information into mathematical equations, then employed algebraic methods to solve for the unknowns, and finally evaluated the function at the desired point. This process exemplifies the essence of mathematical problem-solving, which involves translating real-world scenarios into abstract models, manipulating these models to extract information, and then interpreting the results in the context of the original problem. The skills we have honed in this exercise are not limited to exponential functions but are applicable to a wide range of mathematical challenges. The ability to think critically, analyze information, and apply appropriate techniques is a valuable asset in any field. Thus, the solution to this problem is not just a numerical answer but also a demonstration of our problem-solving prowess and our understanding of fundamental mathematical principles.