Evaluate The Function $f(x)=\log _5 X$.A. Find $f\left(\frac{1}{5}\right$\].

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Introduction

In this article, we will evaluate the function $f(x)=\log _5 x$ and find its value at a specific point. The function $f(x)=\log _5 x$ is a logarithmic function with base 5. It is defined for all positive real numbers and has a range of all real numbers.

The Function $f(x)=\log _5 x$

The function $f(x)=\log _5 x$ is defined as the exponent to which the base 5 must be raised to produce the number $x$. In other words, if $y = \log _5 x$, then $5^y = x$. This function is a one-to-one function, which means that it passes the horizontal line test. It is also a continuous function, which means that it can be graphed without lifting the pencil from the paper.

Finding $f\left(\frac{1}{5}\right)$

To find $f\left(\frac{1}{5}\right)$, we need to find the value of $x$ such that $f(x) = \log _5 \left(\frac{1}{5}\right)$. We can rewrite $\frac{1}{5}$ as $5^{-1}$, since $5^{-1} = \frac{1}{5}$. Therefore, we have:

$$f\left(\frac{1}{5}\right) = \log _5 \left(5^{-1}\right)$$

Using the property of logarithms that $\log _a a^x = x$, we can rewrite the above equation as:

$$f\left(\frac{1}{5}\right) = -1$$

Therefore, the value of $f\left(\frac{1}{5}\right)$ is $-1$.

Properties of the Function $f(x)=\log _5 x$

The function $f(x)=\log _5 x$ has several important properties that we need to know. These properties include:

• Domain: The domain of the function $f(x)=\log _5 x$ is all positive real numbers.
• Range: The range of the function $f(x)=\log _5 x$ is all real numbers.
• One-to-one: The function $f(x)=\log _5 x$ is a one-to-one function, which means that it passes the horizontal line test.
• Continuous: The function $f(x)=\log _5 x$ is a continuous function, which means that it can be graphed without lifting the pencil from the paper.
• Invertible: The function $f(x)=\log _5 x$ is invertible, which means that it has an inverse function.

Graph of the Function $f(x)=\log _5 x$

The graph of the function $f(x)=\log _5 x$ is a logarithmic curve that opens upwards. It has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=-\infty$. The graph of the function $f(x)=\log _5 x$ is shown below:

Conclusion

In this article, we evaluated the function $f(x)=\log _5 x$ and found its value at a specific point. We also discussed the properties of the function $f(x)=\log _5 x$, including its domain, range, one-to-one property, continuous property, and invertible property. Finally, we graphed the function $f(x)=\log _5 x$ and discussed its behavior.

References

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Properties of Logarithmic Functions" by Purplemath
• [3] "Graphing Logarithmic Functions" by Math Is Fun

Further Reading

If you want to learn more about logarithmic functions, I recommend checking out the following resources:

• "Logarithmic Functions" by Khan Academy
• "Properties of Logarithmic Functions" by IXL
• "Graphing Logarithmic Functions" by Wolfram Alpha

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Q&A

Q: What is the domain of the function $f(x)=\log _5 x$?

A: The domain of the function $f(x)=\log _5 x$ is all positive real numbers.

Q: What is the range of the function $f(x)=\log _5 x$?

A: The range of the function $f(x)=\log _5 x$ is all real numbers.

Q: Is the function $f(x)=\log _5 x$ one-to-one?

A: Yes, the function $f(x)=\log _5 x$ is a one-to-one function, which means that it passes the horizontal line test.

Q: Is the function $f(x)=\log _5 x$ continuous?

A: Yes, the function $f(x)=\log _5 x$ is a continuous function, which means that it can be graphed without lifting the pencil from the paper.

Q: Is the function $f(x)=\log _5 x$ invertible?

A: Yes, the function $f(x)=\log _5 x$ is invertible, which means that it has an inverse function.

Q: How do you find the value of $f\left(\frac{1}{5}\right)$?

A: To find the value of $f\left(\frac{1}{5}\right)$, we need to find the value of $x$ such that $f(x) = \log _5 \left(\frac{1}{5}\right)$. We can rewrite $\frac{1}{5}$ as $5^{-1}$, since $5^{-1} = \frac{1}{5}$. Therefore, we have:

$$f\left(\frac{1}{5}\right) = \log _5 \left(5^{-1}\right)$$

Using the property of logarithms that $\log _a a^x = x$, we can rewrite the above equation as:

$$f\left(\frac{1}{5}\right) = -1$$

Therefore, the value of $f\left(\frac{1}{5}\right)$ is $-1$.

Q: What is the graph of the function $f(x)=\log _5 x$?

A: The graph of the function $f(x)=\log _5 x$ is a logarithmic curve that opens upwards. It has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=-\infty$.

Q: What are some important properties of the function $f(x)=\log _5 x$?

A: Some important properties of the function $f(x)=\log _5 x$ include:

• Domain: The domain of the function $f(x)=\log _5 x$ is all positive real numbers.
• Range: The range of the function $f(x)=\log _5 x$ is all real numbers.
• One-to-one: The function $f(x)=\log _5 x$ is a one-to-one function, which means that it passes the horizontal line test.
• Continuous: The function $f(x)=\log _5 x$ is a continuous function, which means that it can be graphed without lifting the pencil from the paper.
• Invertible: The function $f(x)=\log _5 x$ is invertible, which means that it has an inverse function.

Q: How do you graph the function $f(x)=\log _5 x$?

A: To graph the function $f(x)=\log _5 x$, you can use a graphing calculator or a computer program. You can also use a table of values to graph the function.

Q: What are some real-world applications of the function $f(x)=\log _5 x$?

A: Some real-world applications of the function $f(x)=\log _5 x$ include:

• Sound levels: The decibel scale is based on the logarithmic function $f(x)=\log _5 x$.
• Light levels: The lux scale is based on the logarithmic function $f(x)=\log _5 x$.
• Chemical reactions: The rate of a chemical reaction can be modeled using the logarithmic function $f(x)=\log _5 x$.

Conclusion

In this article, we evaluated the function $f(x)=\log _5 x$ and answered some common questions about the function. We also discussed the properties of the function $f(x)=\log _5 x$, including its domain, range, one-to-one property, continuous property, and invertible property. Finally, we graphed the function $f(x)=\log _5 x$ and discussed its behavior.

References

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Properties of Logarithmic Functions" by Purplemath
• [3] "Graphing Logarithmic Functions" by Math Is Fun

Further Reading

If you want to learn more about logarithmic functions, I recommend checking out the following resources:

• "Logarithmic Functions" by Khan Academy
• "Properties of Logarithmic Functions" by IXL
• "Graphing Logarithmic Functions" by Wolfram Alpha