An Exponential Function Of The Form $f(x) = B^x$ Includes The Points \[$(2, 16)\$\], \[$(3, 64)\$\], And \[$(4, 256)\$\]. What Is The Value Of $b$?

Introduction

Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and economics. An exponential function of the form $f(x) = b^x$ is a function that grows or decays exponentially with respect to the input variable $x$. In this article, we will discuss how to find the value of $b$ in an exponential function given three points on the graph of the function.

Understanding Exponential Functions

An exponential function of the form $f(x) = b^x$ is a function that can be written in the form $y = ab^x$, where $a$ and $b$ are constants. The base $b$ is a positive real number, and the exponent $x$ is a real number. The graph of an exponential function is a curve that passes through the point $(0, a)$ and has a horizontal asymptote at $y = 0$ if $b < 1$ or $y = \infty$ if $b > 1$.

Given Points on the Graph

We are given three points on the graph of the exponential function $f(x) = b^x$: $(2, 16)$, $(3, 64)$, and $(4, 256)$. These points lie on the graph of the function, and we can use them to find the value of $b$.

Using the Points to Find the Value of b

To find the value of $b$, we can use the fact that the points $(2, 16)$, $(3, 64)$, and $(4, 256)$ lie on the graph of the function $f(x) = b^x$. We can write the following equations using the given points:

$$b^2 = 16$$

$$b^3 = 64$$

$$b^4 = 256$$

Solving for b

We can solve for $b$ by taking the square root of both sides of the first equation, the cube root of both sides of the second equation, and the fourth root of both sides of the third equation.

$$b = \sqrt{16} = \pm 4$$

$$b = \sqrt[3]{64} = \pm 4$$

$$b = \sqrt[4]{256} = \pm 4$$

Conclusion

Since the value of $b$ must be positive, we can conclude that $b = 4$. Therefore, the value of $b$ in the exponential function $f(x) = b^x$ is $4$.

Example Use Case

The value of $b$ in an exponential function can be used to model real-world phenomena, such as population growth or radioactive decay. For example, if we know that a population grows exponentially with a base of $4$, we can use this value to model the population growth over time.

Conclusion

In conclusion, we have discussed how to find the value of $b$ in an exponential function given three points on the graph of the function. We have used the fact that the points lie on the graph of the function to write equations and solve for $b$. We have also discussed an example use case of the value of $b$ in an exponential function.

Final Answer

The final answer is $\boxed{4}$.

Introduction

In our previous article, we discussed how to find the value of $b$ in an exponential function given three points on the graph of the function. In this article, we will answer some frequently asked questions related to exponential functions and the value of $b$.

Q: What is an exponential function?

A: An exponential function is a function of the form $f(x) = b^x$, where $b$ is a positive real number and $x$ is a real number. The graph of an exponential function is a curve that passes through the point $(0, a)$ and has a horizontal asymptote at $y = 0$ if $b < 1$ or $y = \infty$ if $b > 1$.

Q: How do I find the value of $b$ in an exponential function?

A: To find the value of $b$ in an exponential function, you can use the fact that the points $(2, 16)$, $(3, 64)$, and $(4, 256)$ lie on the graph of the function $f(x) = b^x$. You can write the following equations using the given points:

$$b^2 = 16$$

$$b^3 = 64$$

$$b^4 = 256$$

You can then solve for $b$ by taking the square root of both sides of the first equation, the cube root of both sides of the second equation, and the fourth root of both sides of the third equation.

Q: What is the significance of the value of $b$ in an exponential function?

A: The value of $b$ in an exponential function is significant because it determines the rate of growth or decay of the function. If $b > 1$, the function grows exponentially, and if $b < 1$, the function decays exponentially.

Q: Can I use the value of $b$ to model real-world phenomena?

A: Yes, you can use the value of $b$ to model real-world phenomena, such as population growth or radioactive decay. For example, if you know that a population grows exponentially with a base of $4$, you can use this value to model the population growth over time.

Q: What are some common applications of exponential functions?

A: Exponential functions have numerous applications in various fields, including science, engineering, and economics. Some common applications of exponential functions include:

• Modeling population growth or decay
• Modeling radioactive decay
• Modeling chemical reactions
• Modeling financial growth or decay
• Modeling the spread of diseases

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the points on a coordinate plane.

Q: What are some common mistakes to avoid when working with exponential functions?

A: Some common mistakes to avoid when working with exponential functions include:

• Assuming that the function is linear when it is actually exponential
• Failing to check the domain and range of the function
• Failing to check the value of $b$ before graphing the function
• Failing to use a graphing calculator or computer program to graph the function

Conclusion

In conclusion, we have answered some frequently asked questions related to exponential functions and the value of $b$. We hope that this article has been helpful in clarifying some of the concepts and applications of exponential functions.

Final Answer

The final answer is $\boxed{4}$.