What Is The Range Of Y = Log ⁡ 8 X Y = \log_8 X Y = Lo G 8 ​ X ?A. All Real Numbers Less Than 0 B. All Real Numbers Greater Than 0 C. All Real Numbers Not Equal To 0 D. All Real Numbers

**What is the Range of $y = \log_8 x$?**

Understanding the Basics of Logarithmic Functions

Before we dive into the specifics of the range of $y = \log_8 x$, let's take a step back and understand the basics of logarithmic functions. A logarithmic function is the inverse of an exponential function. In other words, if we have an exponential function of the form $y = a^x$, where $a$ is a positive real number not equal to 1, then the logarithmic function of the form $y = \log_a x$ is its inverse.

The Domain and Range of Logarithmic Functions

The domain of a logarithmic function is all positive real numbers, while the range is all real numbers. This is because the logarithmic function is the inverse of the exponential function, and the exponential function has a range of all positive real numbers.

The Specific Case of $y = \log_8 x$

Now, let's focus on the specific case of $y = \log_8 x$. This is a logarithmic function with base 8. To find the range of this function, we need to consider the possible values of $x$ and how they affect the value of $y$.

The Range of $y = \log_8 x$

The range of a logarithmic function is all real numbers. This is because the logarithmic function is the inverse of the exponential function, and the exponential function has a range of all positive real numbers.

However, we need to consider the fact that the base of the logarithmic function is 8, which is a positive real number not equal to 1. This means that the range of $y = \log_8 x$ is all real numbers, but we need to exclude the value 0.

Why 0 is Excluded from the Range

The value 0 is excluded from the range of $y = \log_8 x$ because the logarithmic function is undefined when the base is 0. In other words, $\log_8 0$ is undefined.

Conclusion

In conclusion, the range of $y = \log_8 x$ is all real numbers, but we need to exclude the value 0. This is because the logarithmic function is the inverse of the exponential function, and the exponential function has a range of all positive real numbers.

Frequently Asked Questions

Q: What is the range of $y = \log_8 x$?

A: The range of $y = \log_8 x$ is all real numbers, but we need to exclude the value 0.

Q: Why is 0 excluded from the range?

A: The value 0 is excluded from the range of $y = \log_8 x$ because the logarithmic function is undefined when the base is 0.

Q: What is the domain of $y = \log_8 x$?

A: The domain of $y = \log_8 x$ is all positive real numbers.

Q: What is the base of the logarithmic function $y = \log_8 x$?

A: The base of the logarithmic function $y = \log_8 x$ is 8.

Q: Is the range of $y = \log_8 x$ all real numbers?

A: Yes, the range of $y = \log_8 x$ is all real numbers, but we need to exclude the value 0.

Q: Why is the range of $y = \log_8 x$ all real numbers?

A: The range of $y = \log_8 x$ is all real numbers because the logarithmic function is the inverse of the exponential function, and the exponential function has a range of all positive real numbers.

Q: What is the inverse of the exponential function $y = a^x$?

A: The inverse of the exponential function $y = a^x$ is the logarithmic function $y = \log_a x$.

Q: What is the domain of the logarithmic function $y = \log_a x$?

A: The domain of the logarithmic function $y = \log_a x$ is all positive real numbers.

Q: What is the range of the logarithmic function $y = \log_a x$?

A: The range of the logarithmic function $y = \log_a x$ is all real numbers.

Q: Why is the range of the logarithmic function $y = \log_a x$ all real numbers?

A: The range of the logarithmic function $y = \log_a x$ is all real numbers because the logarithmic function is the inverse of the exponential function, and the exponential function has a range of all positive real numbers.