What Is The Equation Of The Vertical Asymptote Of $h(x)=6 \log _2(x-3)-5$?Enter Your Answer In The Box. X = X= X = □ \square □

Introduction

In mathematics, a vertical asymptote is a vertical line that a function approaches but never touches. It is an important concept in calculus and is used to determine the behavior of a function as it approaches a certain point. In this article, we will discuss the equation of the vertical asymptote of a logarithmic function.

What is a Logarithmic Function?

A logarithmic function is a function that is the inverse of an exponential function. It is a function that takes a positive real number as input and returns the exponent to which a base number must be raised to produce the input number. The general form of a logarithmic function is:

$$f(x) = \log_b(x)$$

where $b$ is the base of the logarithm and $x$ is the input.

The Equation of the Vertical Asymptote

The equation of the vertical asymptote of a logarithmic function is given by:

$$x = a$$

where $a$ is the value of the input that makes the function undefined.

Finding the Equation of the Vertical Asymptote

To find the equation of the vertical asymptote of a logarithmic function, we need to find the value of the input that makes the function undefined. This value is called the asymptote.

For the function $h(x) = 6 \log_2(x-3) - 5$, we need to find the value of $x$ that makes the function undefined.

Step 1: Identify the Base of the Logarithm

The base of the logarithm is 2.

Step 2: Identify the Input

The input is $x-3$.

Step 3: Find the Value of the Input that Makes the Function Undefined

The function is undefined when the input is less than or equal to 0. Therefore, we need to find the value of $x$ that makes $x-3 \leq 0$.

Step 4: Solve the Inequality

To solve the inequality $x-3 \leq 0$, we need to add 3 to both sides of the inequality.

$$x-3+3 \leq 0+3$$

$$x \leq 3$$

Step 5: Write the Equation of the Vertical Asymptote

The equation of the vertical asymptote is given by:

$$x = 3$$

Conclusion

In this article, we discussed the equation of the vertical asymptote of a logarithmic function. We found that the equation of the vertical asymptote is given by $x = a$, where $a$ is the value of the input that makes the function undefined. We also found that the equation of the vertical asymptote of the function $h(x) = 6 \log_2(x-3) - 5$ is $x = 3$.

Example

Find the equation of the vertical asymptote of the function $f(x) = 4 \log_3(x-2) + 2$.

Solution

To find the equation of the vertical asymptote, we need to follow the same steps as before.

Step 1: Identify the Base of the Logarithm

The base of the logarithm is 3.

Step 2: Identify the Input

The input is $x-2$.

Step 3: Find the Value of the Input that Makes the Function Undefined

The function is undefined when the input is less than or equal to 0. Therefore, we need to find the value of $x$ that makes $x-2 \leq 0$.

Step 4: Solve the Inequality

To solve the inequality $x-2 \leq 0$, we need to add 2 to both sides of the inequality.

$$x-2+2 \leq 0+2$$

$$x \leq 2$$

Step 5: Write the Equation of the Vertical Asymptote

The equation of the vertical asymptote is given by:

$$x = 2$$

Final Answer

$$

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that a function approaches but never touches. It is an important concept in calculus and is used to determine the behavior of a function as it approaches a certain point.

Q: How do I find the equation of the vertical asymptote of a function?

A: To find the equation of the vertical asymptote of a function, you need to follow these steps:

1. Identify the base of the logarithm.
2. Identify the input.
3. Find the value of the input that makes the function undefined.
4. Solve the inequality to find the value of the input that makes the function undefined.
5. Write the equation of the vertical asymptote.

Q: What is the equation of the vertical asymptote of the function $h(x) = 6 \log_2(x-3) - 5$?

A: The equation of the vertical asymptote of the function $h(x) = 6 \log_2(x-3) - 5$ is $x = 3$.

Q: How do I know if a function has a vertical asymptote?

A: A function has a vertical asymptote if the input is less than or equal to 0. This means that the function is undefined when the input is less than or equal to 0.

Q: Can a function have more than one vertical asymptote?

A: Yes, a function can have more than one vertical asymptote. This occurs when the function has multiple inputs that make the function undefined.

Q: How do I graph a function with a vertical asymptote?

A: To graph a function with a vertical asymptote, you need to draw a vertical line at the point where the function is undefined. The function will approach this line but never touch it.

Q: What is the difference between a vertical asymptote and a hole in a graph?

A: A vertical asymptote is a vertical line that a function approaches but never touches. A hole in a graph is a point where the function is undefined, but the function approaches this point and touches it.

Q: Can a function have a hole in its graph and a vertical asymptote?

A: Yes, a function can have a hole in its graph and a vertical asymptote. This occurs when the function has multiple inputs that make the function undefined.

Q: How do I find the equation of the vertical asymptote of a rational function?

A: To find the equation of the vertical asymptote of a rational function, you need to follow these steps:

1. Factor the numerator and denominator of the function.
2. Identify the values of the input that make the denominator equal to 0.
3. Write the equation of the vertical asymptote.

Q: What is the equation of the vertical asymptote of the function $f(x) = \frac{1}{x-2}$?

A: The equation of the vertical asymptote of the function $f(x) = \frac{1}{x-2}$ is $x = 2$.

Q: Can a function have a vertical asymptote and a horizontal asymptote?

A: Yes, a function can have a vertical asymptote and a horizontal asymptote. This occurs when the function has multiple inputs that make the function undefined and the function approaches a certain value as the input approaches infinity.

Q: How do I find the equation of the horizontal asymptote of a function?

A: To find the equation of the horizontal asymptote of a function, you need to follow these steps:

1. Identify the degree of the numerator and denominator of the function.
2. Compare the degrees of the numerator and denominator.
3. Write the equation of the horizontal asymptote.

Q: What is the equation of the horizontal asymptote of the function $f(x) = \frac{1}{x}$?

A: The equation of the horizontal asymptote of the function $f(x) = \frac{1}{x}$ is $y = 0$.

Final Answer

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