The Function F F F Is Given By F ( X ) = Log ⁡ 2 X + Log ⁡ 2 ( X − 2 F(x)=\log _2 X+\log _2(x-2 F ( X ) = Lo G 2 ​ X + Lo G 2 ​ ( X − 2 ]. What Are All Values Of X X X For Which F ( X ) = 3 F(x)=3 F ( X ) = 3 ?A. X = 3 X=3 X = 3 OnlyB. X = 4 X=4 X = 4 OnlyC. X = − 2 X=-2 X = − 2 And X = 4 X=4 X = 4 D. X = − 1 X=-1 X = − 1 And

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Introduction

In mathematics, functions play a crucial role in understanding various mathematical concepts and their applications. The function $f$ given by $f(x)=\log _2 x+\log _2(x-2)$ is a logarithmic function that involves the sum of two logarithms with base 2. In this article, we will explore the values of $x$ for which $f(x)=3$. To do this, we will first analyze the function and then solve the equation $f(x)=3$ to find the required values of $x$.

Understanding the Function $f$

The function $f$ is defined as $f(x)=\log _2 x+\log _2(x-2)$. This function involves the sum of two logarithms with base 2. To simplify the function, we can use the logarithmic property that states $\log _a b + \log _a c = \log _a (bc)$. Applying this property to the function $f$, we get:

$$f(x)=\log _2 x+\log _2(x-2)=\log _2 (x(x-2))$$

This simplified form of the function $f$ makes it easier to analyze and solve the equation $f(x)=3$.

Solving the Equation $f(x)=3$

To solve the equation $f(x)=3$, we substitute the simplified form of the function $f$ into the equation:

$$\log _2 (x(x-2))=3$$

Using the definition of a logarithm, we can rewrite the equation as:

$$x(x-2)=2^3$$

Simplifying the equation further, we get:

$$x(x-2)=8$$

Expanding the left-hand side of the equation, we get:

$$x^2-2x-8=0$$

This is a quadratic equation in $x$. To solve for $x$, we can use the quadratic formula:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

In this case, $a=1$, $b=-2$, and $c=-8$. Substituting these values into the quadratic formula, we get:

$$x=\frac{-(-2)\pm\sqrt{(-2)^2-4(1)(-8)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x=\frac{2\pm\sqrt{4+32}}{2}$$

$$x=\frac{2\pm\sqrt{36}}{2}$$

$$x=\frac{2\pm6}{2}$$

This gives us two possible solutions for $x$:

$$x=\frac{2+6}{2}=4$$

$$x=\frac{2-6}{2}=-2$$

Conclusion

In this article, we explored the function $f$ given by $f(x)=\log _2 x+\log _2(x-2)$ and solved the equation $f(x)=3$ to find the values of $x$ for which the function is equal to 3. We simplified the function using logarithmic properties and then solved the resulting quadratic equation to find the required values of $x$. The solutions to the equation are $x=4$ and $x=-2$. Therefore, the correct answer is:

The final answer is C. $x=-2$ and $x=4$

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Introduction

In our previous article, we explored the function $f$ given by $f(x)=\log _2 x+\log _2(x-2)$ and solved the equation $f(x)=3$ to find the values of $x$ for which the function is equal to 3. In this article, we will answer some frequently asked questions related to the function $f$ and its solutions.

Q&A

Q1: What is the domain of the function $f$?

A1: The domain of the function $f$ is all real numbers $x$ such that $x>0$ and $x>2$. This is because the logarithmic function is defined only for positive real numbers, and the expression $x-2$ must also be positive.

Q2: How do you simplify the function $f$?

A2: The function $f$ can be simplified using the logarithmic property that states $\log _a b + \log _a c = \log _a (bc)$. Applying this property to the function $f$, we get:

$$f(x)=\log _2 x+\log _2(x-2)=\log _2 (x(x-2))$$

Q3: How do you solve the equation $f(x)=3$?

A3: To solve the equation $f(x)=3$, we substitute the simplified form of the function $f$ into the equation:

$$\log _2 (x(x-2))=3$$

Using the definition of a logarithm, we can rewrite the equation as:

$$x(x-2)=2^3$$

Simplifying the equation further, we get:

$$x(x-2)=8$$

Expanding the left-hand side of the equation, we get:

$$x^2-2x-8=0$$

This is a quadratic equation in $x$. To solve for $x$, we can use the quadratic formula:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Q4: What are the solutions to the equation $f(x)=3$?

A4: The solutions to the equation $f(x)=3$ are $x=4$ and $x=-2$.

Q5: Why are there two solutions to the equation $f(x)=3$?

A5: There are two solutions to the equation $f(x)=3$ because the quadratic equation $x^2-2x-8=0$ has two distinct roots. This is due to the fact that the discriminant $b^2-4ac$ is positive, which means that the quadratic equation has two real and distinct solutions.

Q6: Can you graph the function $f$?

A6: Yes, we can graph the function $f$ using a graphing calculator or a computer algebra system. The graph of the function $f$ is a logarithmic curve that has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=-\infty$.

Q7: What is the range of the function $f$?

A7: The range of the function $f$ is all real numbers $y$ such that $y>-\infty$. This is because the logarithmic function is defined only for positive real numbers, and the expression $x-2$ must also be positive.

Conclusion

In this article, we answered some frequently asked questions related to the function $f$ and its solutions. We hope that this Q&A article has provided you with a better understanding of the function $f$ and its properties. If you have any further questions, please don't hesitate to ask.

The final answer is C. $x=-2$ and $x=4$