Forming the System of Equations

Oct 13, 2025 by windailys.com 51 views

Introduction

Logarithmic 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 a specific logarithmic function of the form $y = a + b \ln(x + 4)$ , where $a$ and $b$ are constants that need to be determined. We will use the given points $(-3, -3)$ and $(655, 34)$ to find the values of $a$ and $b$ , rounding all values to three decimal places if necessary.

The Logarithmic Function

The given logarithmic function is of the form $y = a + b \ln(x + 4)$ . This function is a combination of a linear function and a logarithmic function. The linear function is represented by $a$ , while the logarithmic function is represented by $b \ln(x + 4)$ . The logarithmic function is a fundamental concept in mathematics, and it is used to model various phenomena in real-world applications.

The Given Points

We are given two points, $(-3, -3)$ and $(655, 34)$ , that lie on the logarithmic function. These points can be used to form a system of equations, which can be solved to find the values of $a$ and $b$ .

Forming the System of Equations

To form the system of equations, we will substitute the given points into the logarithmic function. For the point $(-3, -3)$ , we have:

-3 = a + b \ln(-3 + 4)

Simplifying the equation, we get:

-3 = a + b \ln(1)

Since $\ln(1) = 0$ , the equation becomes:

-3 = a

For the point $(655, 34)$ , we have:

34 = a + b \ln(655 + 4)

Simplifying the equation, we get:

34 = a + b \ln(659)

Solving the System of Equations

We now have a system of two equations with two unknowns:

-3 = a

34 = a + b \ln(659)

Substituting the value of $a$ from the first equation into the second equation, we get:

34 = -3 + b \ln(659)

Simplifying the equation, we get:

37 = b \ln(659)

Dividing both sides by $\ln(659)$ , we get:

b = \frac{37}{\ln(659)}

Using a calculator to evaluate the expression, we get:

b \approx 1.061

Finding the Value of $a$

We already found the value of $a$ in the previous section:

a = -3

Conclusion

In this article, we used the given points $(-3, -3)$ and $(655, 34)$ to find the values of $a$ and $b$ in the logarithmic function $y = a + b \ln(x + 4)$ . We formed a system of equations using the given points and solved for the values of $a$ and $b$ . We found that $a \approx -3$ and $b \approx 1.061$ , rounding all values to three decimal places if necessary.

Final Answer

The final answer is:

$a \approx -3$

$b \approx 1.061$

References

[1] "Logarithmic Functions". Math Open Reference. Retrieved 2023-02-26.
[2] "Logarithmic Functions". Wolfram MathWorld. Retrieved 2023-02-26.

Additional Resources

[1] "Logarithmic Functions". Khan Academy. Retrieved 2023-02-26.
[2] "Logarithmic Functions". MIT OpenCourseWare. Retrieved 2023-02-26.

Introduction

In our previous article, we discussed a logarithmic function of the form $y = a + b \ln(x + 4)$ and found the values of $a$ and $b$ using the given points $(-3, -3)$ and $(655, 34)$ . In this article, we will answer some frequently asked questions related to logarithmic functions and provide additional information to help readers better understand the concept.

Q&A

Q: What is a logarithmic function?

A: A logarithmic function is a mathematical function that is the inverse of an exponential function. It is used to model various phenomena in real-world applications, such as population growth, chemical reactions, and financial transactions.

Q: What is the general form of a logarithmic function?

A: The general form of a logarithmic function is $y = a + b \ln(x + c)$ , where $a$ and $b$ are constants, and $c$ is a constant that shifts the logarithmic function horizontally.

Q: How do I find the values of $a$ and $b$ in a logarithmic function?

A: To find the values of $a$ and $b$ in a logarithmic function, you need to use the given points that lie on the function. You can form a system of equations using the given points and solve for the values of $a$ and $b$ .

Q: What is the significance of the natural logarithm in a logarithmic function?

A: The natural logarithm is a fundamental concept in mathematics, and it is used to model various phenomena in real-world applications. In a logarithmic function, the natural logarithm is used to represent the growth or decay of a quantity over time.

Q: Can I use a logarithmic function to model a non-linear relationship?

A: Yes, you can use a logarithmic function to model a non-linear relationship. Logarithmic functions are used to model various phenomena in real-world applications, such as population growth, chemical reactions, and financial transactions.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you need to use a graphing calculator or a computer software. You can also use a table of values to graph the function.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all real numbers except for the value of $x$ that makes the argument of the logarithm equal to zero.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function is all real numbers.

Additional Information

Logarithmic functions are used to model various phenomena in real-world applications, such as population growth, chemical reactions, and financial transactions.
Logarithmic functions are used to represent the growth or decay of a quantity over time.
Logarithmic functions are used to model non-linear relationships.
Logarithmic functions are used to graph various phenomena in real-world applications.

Conclusion

In this article, we answered some frequently asked questions related to logarithmic functions and provided additional information to help readers better understand the concept. We hope that this article has been helpful in providing a better understanding of logarithmic functions and their applications.

Final Answer

The final answer is:

A logarithmic function is a mathematical function that is the inverse of an exponential function.
The general form of a logarithmic function is $y = a + b \ln(x + c)$ .
To find the values of $a$ and $b$ in a logarithmic function, you need to use the given points that lie on the function.
The natural logarithm is a fundamental concept in mathematics, and it is used to model various phenomena in real-world applications.
Logarithmic functions are used to model non-linear relationships.
The domain of a logarithmic function is all real numbers except for the value of $x$ that makes the argument of the logarithm equal to zero.
The range of a logarithmic function is all real numbers.

References

[1] "Logarithmic Functions". Math Open Reference. Retrieved 2023-02-26.
[2] "Logarithmic Functions". Wolfram MathWorld. Retrieved 2023-02-26.

Additional Resources

[1] "Logarithmic Functions". Khan Academy. Retrieved 2023-02-26.
[2] "Logarithmic Functions". MIT OpenCourseWare. Retrieved 2023-02-26.