Complete The Table For The Function Y = Log ⁡ ( X Y=\log (x Y = Lo G ( X ].${ \begin{array}{|c|c|} \hline x & Y \ \hline \frac{1}{100} & -2 \ \hline \frac{1}{10} & -1 \ \hline 1 & 0 \ \hline 10 & 1 \ \hline \end{array} }$Graph The Function $y=\log

Mar 9, 2025 by ADMIN 249 views

$**Completing the Table for the Function $y=\log(x)$**$

Introduction

In mathematics, logarithmic functions are a crucial part of algebra and calculus. The logarithmic function $y=\log(x)$ is defined as the inverse of the exponential function $y=e^x$ . In this article, we will focus on completing the table for the function $y=\log(x)$ and graphing the function.

Understanding the Logarithmic Function

The logarithmic function $y=\log(x)$ is defined as the inverse of the exponential function $y=e^x$ . This means that if $y=e^x$ , then $x=\log(y)$ . The logarithmic function has a base, which is usually 10 or $e$ . In this case, we are dealing with the natural logarithm, which has a base of $e$ .

Completing the Table

The table provided shows the values of $x$ and $y$ for the function $y=\log(x)$ . However, the table is incomplete, and we need to complete it.

$x$	$y$
$\frac{1}{100}$	$-2$
$\frac{1}{10}$	$-1$
$1$	$0$
$10$	$1$

To complete the table, we need to find the values of $y$ for different values of $x$ . We can use a calculator or a graphing tool to find the values of $y$ .

$x$	$y$
$\frac{1}{100}$	$-2$
$\frac{1}{10}$	$-1$
$1$	$0$
$10$	$1$
$100$	$2$
$1000$	$3$
$10000$	$4$
$100000$	$5$

Graphing the Function

To graph the function $y=\log(x)$ , we can use a graphing tool or a calculator. The graph of the function will be a curve that approaches the x-axis as $x$ approaches 0.

Properties of the Logarithmic Function

The logarithmic function has several properties that are important to understand.

Domain: The domain of the logarithmic function is all positive real numbers.
Range: The range of the logarithmic function is all real numbers.
Asymptote: The asymptote of the logarithmic function is the x-axis.
Increasing/Decreasing: The logarithmic function is increasing for all positive values of $x$ .

Real-World Applications

The logarithmic function has several real-world applications.

Finance: The logarithmic function is used in finance to calculate the return on investment.
Science: The logarithmic function is used in science to calculate the pH of a solution.
Engineering: The logarithmic function is used in engineering to calculate the decibel level of a sound.

Conclusion

In conclusion, the logarithmic function $y=\log(x)$ is an important function in mathematics. Completing the table for the function and graphing the function are essential skills to understand the properties of the function. The logarithmic function has several real-world applications, and it is used in finance, science, and engineering.

References

[1] "Logarithmic Functions" by Math Open Reference
[2] "Logarithmic Functions" by Khan Academy
[3] "Logarithmic Functions" by Wolfram MathWorld

Additional Resources

[1] "Logarithmic Functions" by MIT OpenCourseWare
[2] "Logarithmic Functions" by University of California, Berkeley
[3] "Logarithmic Functions" by University of Michigan
Logarithmic Function Q&A ==========================

Introduction

In our previous article, we discussed the logarithmic function $y=\log(x)$ and its properties. In this article, we will answer some frequently asked questions about the logarithmic function.

Q: What is the domain of the logarithmic function?

A: The domain of the logarithmic function is all positive real numbers. This means that the input of the function must be greater than 0.

Q: What is the range of the logarithmic function?

A: The range of the logarithmic function is all real numbers. This means that the output of the function can be any real number.

Q: What is the asymptote of the logarithmic function?

A: The asymptote of the logarithmic function is the x-axis. This means that as the input of the function approaches 0, the output of the function approaches negative infinity.

Q: Is the logarithmic function increasing or decreasing?

A: The logarithmic function is increasing for all positive values of $x$ . This means that as the input of the function increases, the output of the function also increases.

Q: How do I graph the logarithmic function?

A: To graph the logarithmic function, you can use a graphing tool or a calculator. The graph of the function will be a curve that approaches the x-axis as $x$ approaches 0.

Q: What are some real-world applications of the logarithmic function?

A: The logarithmic function has several real-world applications, including:

Finance: The logarithmic function is used in finance to calculate the return on investment.
Science: The logarithmic function is used in science to calculate the pH of a solution.
Engineering: The logarithmic function is used in engineering to calculate the decibel level of a sound.

Q: How do I calculate the logarithm of a number?

A: To calculate the logarithm of a number, you can use a calculator or a graphing tool. You can also use the change of base formula to calculate the logarithm of a number.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows you to calculate the logarithm of a number in a different base. The formula is:

\log_b(a) = \frac{\log_c(a)}{\log_c(b)}

where $a$ is the number, $b$ is the base, and $c$ is the new base.

Q: How do I use the change of base formula?

A: To use the change of base formula, you need to know the logarithm of the number in the new base. You can use a calculator or a graphing tool to find the logarithm of the number in the new base.

Q: What are some common logarithmic functions?

A: Some common logarithmic functions include:

$y=\log(x)$ : This is the natural logarithmic function.
$y=\log_{10}(x)$ : This is the common logarithmic function.
$y=\log_{e}(x)$ : This is the natural logarithmic function.

Conclusion

In conclusion, the logarithmic function $y=\log(x)$ is an important function in mathematics. We have answered some frequently asked questions about the logarithmic function, including its domain, range, asymptote, and real-world applications.

References

[1] "Logarithmic Functions" by Math Open Reference
[2] "Logarithmic Functions" by Khan Academy
[3] "Logarithmic Functions" by Wolfram MathWorld

Additional Resources

[1] "Logarithmic Functions" by MIT OpenCourseWare
[2] "Logarithmic Functions" by University of California, Berkeley
[3] "Logarithmic Functions" by University of Michigan