Use The Function Below To Find F ( 4 F(4 F ( 4 ]. F ( X ) = 5 ⋅ ( 1 3 ) X F(x)=5 \cdot\left(\frac{1}{3}\right)^x F ( X ) = 5 ⋅ ( 3 1 ​ ) X A. 5 81 \frac{5}{81} 81 5 ​ B. 20 3 \frac{20}{3} 3 20 ​ C. 5 12 \frac{5}{12} 12 5 ​ D. 20 81 \frac{20}{81} 81 20 ​

Introduction

Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and economics. In this article, we will focus on solving exponential functions, specifically the function $F(x) = 5 \cdot \left(\frac{1}{3}\right)^x$. We will use this function to find the value of $F(4)$ and explore the properties of exponential functions.

Understanding Exponential Functions

An exponential function is a function of the form $f(x) = a \cdot b^x$, where $a$ and $b$ are constants, and $x$ is the variable. In the function $F(x) = 5 \cdot \left(\frac{1}{3}\right)^x$, $a = 5$ and $b = \frac{1}{3}$. The base of the exponential function is $\frac{1}{3}$, which is a fraction between 0 and 1.

Properties of Exponential Functions

Exponential functions have several important properties that make them useful in mathematics and other fields. Some of the key properties of exponential functions include:

• Exponential growth: Exponential functions grow rapidly as the value of $x$ increases. This is because the base of the exponential function is greater than 1.
• Exponential decay: Exponential functions decay rapidly as the value of $x$ increases. This is because the base of the exponential function is less than 1.
• One-to-one correspondence: Exponential functions are one-to-one, meaning that each value of $x$ corresponds to a unique value of $f(x)$.

Solving Exponential Functions

To solve an exponential function, we need to isolate the variable $x$. In the function $F(x) = 5 \cdot \left(\frac{1}{3}\right)^x$, we can isolate $x$ by taking the logarithm of both sides of the equation.

Using Logarithms to Solve Exponential Functions

Logarithms are a powerful tool for solving exponential functions. By taking the logarithm of both sides of the equation, we can eliminate the exponential term and solve for $x$.

Step 1: Take the Logarithm of Both Sides

To solve the equation $F(x) = 5 \cdot \left(\frac{1}{3}\right)^x$, we can take the logarithm of both sides of the equation. This gives us:

$$\log(F(x)) = \log(5 \cdot \left(\frac{1}{3}\right)^x)$$

Step 2: Use the Logarithm Property

Using the logarithm property $\log(ab) = \log(a) + \log(b)$, we can rewrite the equation as:

$$\log(F(x)) = \log(5) + \log\left(\left(\frac{1}{3}\right)^x\right)$$

Step 3: Simplify the Equation

Using the logarithm property $\log(a^b) = b \cdot \log(a)$, we can simplify the equation as:

$$\log(F(x)) = \log(5) + x \cdot \log\left(\frac{1}{3}\right)$$

Step 4: Isolate $x$

To isolate $x$, we can subtract $\log(5)$ from both sides of the equation and then divide both sides by $\log\left(\frac{1}{3}\right)$:

$$x = \frac{\log(F(x)) - \log(5)}{\log\left(\frac{1}{3}\right)}$$

Finding $F(4)$

Now that we have isolated $x$, we can find the value of $F(4)$ by substituting $x = 4$ into the equation:

$$F(4) = 5 \cdot \left(\frac{1}{3}\right)^4$$

Using a calculator, we can evaluate the expression as:

$$F(4) = 5 \cdot \left(\frac{1}{3}\right)^4 = \frac{5}{81}$$

Conclusion

In this article, we have used the function $F(x) = 5 \cdot \left(\frac{1}{3}\right)^x$ to find the value of $F(4)$. We have also explored the properties of exponential functions and used logarithms to solve the equation. The value of $F(4)$ is $\frac{5}{81}$.

Answer

The correct answer is:

• A. $\frac{5}{81}$

Final Thoughts

Introduction

Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields. In our previous article, we explored the properties of exponential functions and used logarithms to solve equations. In this article, we will answer some frequently asked questions about exponential functions.

Q: What is an exponential function?

A: An exponential function is a function of the form $f(x) = a \cdot b^x$, where $a$ and $b$ are constants, and $x$ is the variable.

Q: What are the properties of exponential functions?

A: Exponential functions have several important properties, including:

• Exponential growth: Exponential functions grow rapidly as the value of $x$ increases.
• Exponential decay: Exponential functions decay rapidly as the value of $x$ increases.
• One-to-one correspondence: Exponential functions are one-to-one, meaning that each value of $x$ corresponds to a unique value of $f(x)$.

Q: How do I solve an exponential function?

A: To solve an exponential function, you can use logarithms to isolate the variable $x$. This involves taking the logarithm of both sides of the equation and then using the logarithm properties to simplify the equation.

Q: What is the difference between exponential growth and exponential decay?

A: Exponential growth occurs when the base of the exponential function is greater than 1, and the value of $x$ increases. Exponential decay occurs when the base of the exponential function is less than 1, and the value of $x$ increases.

Q: How do I use logarithms to solve an exponential function?

A: To use logarithms to solve an exponential function, you can take the logarithm of both sides of the equation and then use the logarithm properties to simplify the equation. This involves using the logarithm property $\log(ab) = \log(a) + \log(b)$ and the logarithm property $\log(a^b) = b \cdot \log(a)$.

Q: What is the value of $F(4)$ for the function $F(x) = 5 \cdot \left(\frac{1}{3}\right)^x$?

A: The value of $F(4)$ for the function $F(x) = 5 \cdot \left(\frac{1}{3}\right)^x$ is $\frac{5}{81}$.

Q: Can I use exponential functions to model real-world phenomena?

A: Yes, exponential functions can be used to model real-world phenomena, such as population growth, chemical reactions, and financial investments.

Q: What are some common applications of exponential functions?

A: Some common applications of exponential functions include:

• Population growth: Exponential functions can be used to model population growth and predict future population sizes.
• Chemical reactions: Exponential functions can be used to model chemical reactions and predict the rate of reaction.
• Financial investments: Exponential functions can be used to model financial investments and predict future returns.

Conclusion

In this article, we have answered some frequently asked questions about exponential functions. We have explored the properties of exponential functions, used logarithms to solve equations, and discussed some common applications of exponential functions. By understanding exponential functions, you can solve complex problems and make predictions about real-world phenomena.

Final Thoughts

Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the properties of exponential functions and using logarithms to solve equations, you can solve complex problems and make predictions about real-world phenomena.