Graph of the Logarithmic Function

Oct 13, 2025 by miodeya.com 45 views

Introduction

Logarithmic functions are a fundamental concept in mathematics, and understanding their behavior is crucial for solving various problems in mathematics and other fields. In this article, we will explore the graph of the function $$y=\log _3(x+2)-1$$ and determine which point corresponds to the real zero of the graph.

What is a Logarithmic Function?

A logarithmic function is a function that is the inverse of an exponential function. It is defined as $$y=\log _a(x)=\frac{\log (x)}{\log (a)}$$ where $a$ is the base of the logarithm. In the given function, the base is 3, and the function is $$y=\log _3(x+2)-1$$.

Graph of the Logarithmic Function

To understand the graph of the function, we need to analyze its behavior. The function is defined as $$y=\log _3(x+2)-1$$, which means that the graph will have a vertical asymptote at $x=-2$ . This is because the logarithm of a negative number is undefined.

The graph of the function will also have a horizontal asymptote at $y=-1$ . This is because as $x$ approaches infinity, the logarithm of $x+2$ approaches infinity, and subtracting 1 from it will result in a horizontal asymptote at $y=-1$ .

Finding the Real Zero of the Graph

The real zero of the graph is the point where the function intersects the x-axis, i.e., where $y=0$ . To find this point, we need to solve the equation $$\log _3(x+2)-1=0$$.

Using the properties of logarithms, we can rewrite the equation as $$\log _3(x+2)=1$$.

Taking the base 3 exponential of both sides, we get $$x+2=3^1$$.

Simplifying the equation, we get $$x+2=3$$.

Subtracting 2 from both sides, we get $$x=1$$.

Therefore, the real zero of the graph is at $x=1$ .

Comparing the Options

Now that we have found the real zero of the graph, we can compare it with the given options.

Option A is $(-1,-1)$ , which is not the real zero of the graph.

Option B is $(1,0)$ , which is the real zero of the graph.

Option C is $(-1,0)$ , which is not the real zero of the graph.

Option D is $(0,-369)$ , which is not the real zero of the graph.

Conclusion

In conclusion, the real zero of the graph of the function $$y=\log _3(x+2)-1$$ is at $x=1$ . This is the point where the function intersects the x-axis, and it is the correct answer among the given options.

Final Answer

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

Additional Information

For those who are interested in learning more about logarithmic functions, here are some additional resources:

References

Logarithmic Functions
Graphing Logarithmic Functions
Solving Logarithmic Equations
Logarithmic Function Q&A ==========================

Frequently Asked Questions

Q: What is a logarithmic function?

A: A logarithmic function is a function that is the inverse of an exponential function. It is defined as $$y=\log _a(x)=\frac{\log (x)}{\log (a)}$$ where $a$ is the base of the logarithm.

Q: What is the graph of a logarithmic function like?

A: The graph of a logarithmic function is a curve that has a vertical asymptote at $x=-2$ and a horizontal asymptote at $y=-1$ . The graph will also have a point of inflection at $x=1$ .

Q: How do I find the real zero of a logarithmic function?

A: To find the real zero of a logarithmic function, you need to solve the equation $$\log _a(x+2)-1=0$$. Using the properties of logarithms, you can rewrite the equation as $$\log _a(x+2)=1$$. Taking the base $a$ exponential of both sides, you get $$x+2=a^1$$. Simplifying the equation, you get $$x+2=a$$. Subtracting 2 from both sides, you get $$x=a-2$$.

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

A: A logarithmic function is the inverse of an exponential function. While an exponential function is defined as $$y=a^x$$, a logarithmic function is defined as $$y=\log _a(x)$$. This means that the graph of a logarithmic function is the reflection of the graph of an exponential function across the line $y=x$ .

Q: Can I use a calculator to find the value of a logarithmic function?

A: Yes, you can use a calculator to find the value of a logarithmic function. Most calculators have a built-in logarithm function that you can use to find the value of a logarithmic expression.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you need to use a graphing calculator or a computer program. You can also use a graphing app on your phone or tablet. To graph a logarithmic function, you need to enter the function into the calculator or app and then adjust the window settings to get a clear view of the graph.

Q: Can I use logarithmic functions in real-world applications?

A: Yes, logarithmic functions have many real-world applications. They are used in fields such as physics, engineering, economics, and computer science. Logarithmic functions are used to model population growth, chemical reactions, and financial transactions.

Q: What are some common logarithmic functions?

A: Some common logarithmic functions include:

$y=\log _2(x)$
$y=\log _3(x)$
$y=\log _4(x)$
$y=\log _5(x)$

Q: Can I use logarithmic functions to solve equations?

A: Yes, logarithmic functions can be used to solve equations. Logarithmic functions are used to solve equations that involve exponential expressions. By using logarithmic functions, you can simplify the equation and solve for the variable.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you need to use the properties of logarithms. The properties of logarithms include:

$\log _a(xy)=\log _a(x)+\log _a(y)$
$\log _a(\frac{x}{y})=\log _a(x)-\log _a(y)$
$\log _a(x^n)=n\log _a(x)$

By using these properties, you can simplify a logarithmic expression and make it easier to solve.

Conclusion

In conclusion, logarithmic functions are an important concept in mathematics. They are used to model real-world phenomena and to solve equations. By understanding logarithmic functions, you can apply them to a wide range of problems and make informed decisions.