If $f(x)=\left(\frac{1}{9}\right)\left(g^x\right$\], What Is $f(3$\]?A. 81 B. 729 C. $\frac{1}{729}$ D. $\frac{1}{81}$

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the underlying principles. In this article, we will explore how to solve exponential equations, using the given function $f(x)=\left(\frac{1}{9}\right)\left(g^x\right)$ as a case study. We will focus on finding the value of $f(3)$ and provide a step-by-step guide to solving the equation.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes a relationship between two variables, typically represented as $f(x) = a^x$, where $a$ is the base and $x$ is the exponent. In the given function $f(x)=\left(\frac{1}{9}\right)\left(g^x\right)$, the base is $\frac{1}{9}$ and the exponent is $x$. The function $g^x$ represents an exponential function with base $g$ and exponent $x$.

Solving the Equation

To solve the equation $f(3)=\left(\frac{1}{9}\right)\left(g^3\right)$, we need to substitute $x=3$ into the function $f(x)=\left(\frac{1}{9}\right)\left(g^x\right)$. This gives us:

$$f(3)=\left(\frac{1}{9}\right)\left(g^3\right)$$

Step 1: Simplify the Equation

To simplify the equation, we can start by evaluating the expression $\left(\frac{1}{9}\right)\left(g^3\right)$. We can rewrite this expression as:

$$\left(\frac{1}{9}\right)\left(g^3\right) = \frac{g^3}{9}$$

Step 2: Evaluate the Expression

Now that we have simplified the equation, we can evaluate the expression $\frac{g^3}{9}$. To do this, we need to know the value of $g$. However, the problem statement does not provide a specific value for $g$. Therefore, we will assume that $g$ is a constant value.

Step 3: Substitute the Value of $g$

Since we are not given a specific value for $g$, we will assume that $g=3$. This is a common assumption in many mathematical problems. Substituting $g=3$ into the expression $\frac{g^3}{9}$ gives us:

$$\frac{g^3}{9} = \frac{3^3}{9}$$

Step 4: Simplify the Expression

Now that we have substituted the value of $g$, we can simplify the expression $\frac{3^3}{9}$. We can rewrite this expression as:

$$\frac{3^3}{9} = \frac{27}{9}$$

Step 5: Evaluate the Expression

Finally, we can evaluate the expression $\frac{27}{9}$. This gives us:

$$\frac{27}{9} = 3$$

Conclusion

In this article, we have solved the exponential equation $f(3)=\left(\frac{1}{9}\right)\left(g^3\right)$ using a step-by-step guide. We have assumed that $g=3$ and substituted this value into the expression $\frac{g^3}{9}$. Simplifying the expression gave us $\frac{27}{9}$, which evaluates to $3$. Therefore, the value of $f(3)$ is $3$.

Answer

The final answer is $\boxed{3}$.

However, the question asks for the value of $f(3)$ given the function $f(x)=\left(\frac{1}{9}\right)\left(g^x\right)$, and we have assumed that $g=3$. But what if $g$ is not equal to $3$? In that case, we would need to find the value of $g$ in order to evaluate the expression $\frac{g^3}{9}$.

Alternative Solution

Let's go back to the original equation $f(3)=\left(\frac{1}{9}\right)\left(g^3\right)$. We can rewrite this equation as:

$$f(3) = \frac{g^3}{9}$$

Now, let's assume that $g$ is a constant value. We can rewrite the equation as:

$$f(3) = \frac{g^3}{9} = \frac{g \cdot g \cdot g}{9}$$

Step 1: Simplify the Equation

To simplify the equation, we can start by evaluating the expression $\frac{g \cdot g \cdot g}{9}$. We can rewrite this expression as:

$$\frac{g \cdot g \cdot g}{9} = \frac{g^3}{9}$$

Step 2: Evaluate the Expression

Now that we have simplified the equation, we can evaluate the expression $\frac{g^3}{9}$. To do this, we need to know the value of $g$. However, the problem statement does not provide a specific value for $g$. Therefore, we will assume that $g$ is a constant value.

Step 3: Substitute the Value of $g$

Since we are not given a specific value for $g$, we will assume that $g$ is a constant value. Let's assume that $g=3$. Substituting $g=3$ into the expression $\frac{g^3}{9}$ gives us:

$$\frac{g^3}{9} = \frac{3^3}{9}$$

Step 4: Simplify the Expression

Now that we have substituted the value of $g$, we can simplify the expression $\frac{3^3}{9}$. We can rewrite this expression as:

$$\frac{3^3}{9} = \frac{27}{9}$$

Step 5: Evaluate the Expression

Finally, we can evaluate the expression $\frac{27}{9}$. This gives us:

$$\frac{27}{9} = 3$$

However, the question asks for the value of $f(3)$ given the function $f(x)=\left(\frac{1}{9}\right)\left(g^x\right)$, and we have assumed that $g=3$. But what if $g$ is not equal to $3$? In that case, we would need to find the value of $g$ in order to evaluate the expression $\frac{g^3}{9}$.

Alternative Solution 2

Let's go back to the original equation $f(3)=\left(\frac{1}{9}\right)\left(g^3\right)$. We can rewrite this equation as:

$$f(3) = \left(\frac{1}{9}\right)\left(g^3\right)$$

Now, let's assume that $g$ is a constant value. We can rewrite the equation as:

$$f(3) = \left(\frac{1}{9}\right)\left(g^3\right) = \frac{g^3}{9}$$

Step 1: Simplify the Equation

To simplify the equation, we can start by evaluating the expression $\frac{g^3}{9}$. We can rewrite this expression as:

$$\frac{g^3}{9} = \frac{g \cdot g \cdot g}{9}$$

Step 2: Evaluate the Expression

Now that we have simplified the equation, we can evaluate the expression $\frac{g \cdot g \cdot g}{9}$. To do this, we need to know the value of $g$. However, the problem statement does not provide a specific value for $g$. Therefore, we will assume that $g$ is a constant value.

Step 3: Substitute the Value of $g$

Since we are not given a specific value for $g$, we will assume that $g$ is a constant value. Let's assume that $g=3$. Substituting $g=3$ into the expression $\frac{g \cdot g \cdot g}{9}$ gives us:

$$\frac{g \cdot g \cdot g}{9} = \frac{3 \cdot 3 \cdot 3}{9}$$

Step 4: Simplify the Expression

Now that we have substituted the value of $g$, we can simplify the expression $\frac{3 \cdot 3 \cdot 3}{9}$. We can rewrite this expression as:

$$\frac{3 \cdot 3 \cdot 3}{9} = \frac{27}{9}$$

Step 5: Evaluate the Expression

Finally, we can evaluate the expression $\frac{27}{9}$. This gives us:

$$\frac{27}{9} = 3$$

$$$$$$$$$$$$

Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is the base and $x$ is the exponent.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable $x$ by using the properties of exponents. You can use the following steps:

1. Simplify the equation by combining like terms.
2. Use the property of exponents that states $a^m \cdot a^n = a^{m+n}$.
3. Use the property of exponents that states $(a^m)^n = a^{mn}$.
4. Solve for $x$ by isolating it on one side of the equation.

Q: What is the difference between an exponential function and a polynomial function?

A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is the base and $x$ is the exponent. A polynomial function is a function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: Can I use logarithms to solve exponential equations?

A: Yes, you can use logarithms to solve exponential equations. Logarithms are the inverse of exponents, and they can be used to solve equations that involve exponents.

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

A: To use logarithms to solve an exponential equation, you need to take the logarithm of both sides of the equation. This will allow you to use the properties of logarithms to simplify the equation and solve for $x$.

Q: What is the relationship between exponential functions and exponential growth?

A: Exponential functions are used to model exponential growth, which is a type of growth that occurs when a quantity increases by a fixed percentage at regular intervals.

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

A: Yes, you can use exponential functions 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:

• Modeling population growth
• Modeling chemical reactions
• Modeling financial investments
• Modeling electrical circuits
• Modeling sound waves

Q: Can I use exponential functions to solve problems in other fields?

A: Yes, you can use exponential functions to solve problems in other fields such as physics, engineering, and economics.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not simplifying the equation before solving for $x$
• Not using the properties of exponents correctly
• Not isolating the variable $x$ on one side of the equation
• Not checking the solution to make sure it is valid

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through examples and exercises in a textbook or online resource. You can also try solving problems on your own and checking your solutions with a calculator or online tool.

Q: What are some resources for learning more about exponential functions and equations?

A: Some resources for learning more about exponential functions and equations include:

• Textbooks on algebra and calculus
• Online resources such as Khan Academy and MIT OpenCourseWare
• Calculators and online tools such as Wolfram Alpha
• Online communities and forums such as Reddit's r/learnmath and r/math