Derivative of a Logarithmic Function

Oct 13, 2025 by miodeya.com 101 views

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Introduction

In mathematics, a function is considered decreasing if its output value decreases as the input value increases. In this article, we will analyze the function $F(x)=\log_{0.75} x$ and determine whether it is decreasing or not.

What is a Logarithmic Function?

A logarithmic function is a mathematical function that is the inverse of an exponential function. It is defined as:

F(x) = \log_b x = y \iff b^y = x

where $b$ is the base of the logarithm, $x$ is the input value, and $y$ is the output value.

The Function $F(x)=\log_{0.75} x$

The function $F(x)=\log_{0.75} x$ is a logarithmic function with base $0.75$ . To determine whether this function is decreasing, we need to analyze its derivative.

Derivative of a Logarithmic Function

The derivative of a logarithmic function is given by:

\frac{d}{dx} \log_b x = \frac{1}{x \ln b}

where $\ln b$ is the natural logarithm of the base $b$ .

Derivative of $F(x)=\log_{0.75} x$

To find the derivative of $F(x)=\log_{0.75} x$ , we substitute $b=0.75$ into the formula:

\frac{d}{dx} \log_{0.75} x = \frac{1}{x \ln 0.75}

Is the Function Decreasing?

To determine whether the function $F(x)=\log_{0.75} x$ is decreasing, we need to analyze the sign of its derivative. If the derivative is negative, then the function is decreasing.

Analysis of the Derivative

The derivative of $F(x)=\log_{0.75} x$ is given by:

\frac{d}{dx} \log_{0.75} x = \frac{1}{x \ln 0.75}

Since $\ln 0.75 < 0$ , the derivative is negative for all $x > 0$ . Therefore, the function $F(x)=\log_{0.75} x$ is decreasing.

Conclusion

In conclusion, the function $F(x)=\log_{0.75} x$ is decreasing because its derivative is negative for all $x > 0$ . This means that as the input value $x$ increases, the output value of the function decreases.

Final Answer

The final answer is:

A. True

The function $F(x)=\log_{0.75} x$ is indeed decreasing.

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Q1: What is the base of the logarithmic function $F(x)=\log_{0.75} x$ ?

A1: The base of the logarithmic function $F(x)=\log_{0.75} x$ is $0.75$ .

Q2: Is the function $F(x)=\log_{0.75} x$ an increasing or decreasing function?

A2: The function $F(x)=\log_{0.75} x$ is a decreasing function.

Q3: Why is the function $F(x)=\log_{0.75} x$ decreasing?

A3: The function $F(x)=\log_{0.75} x$ is decreasing because its derivative is negative for all $x > 0$ . This means that as the input value $x$ increases, the output value of the function decreases.

Q4: What is the derivative of the function $F(x)=\log_{0.75} x$ ?

A4: The derivative of the function $F(x)=\log_{0.75} x$ is given by:

\frac{d}{dx} \log_{0.75} x = \frac{1}{x \ln 0.75}

Q5: Is the function $F(x)=\log_{0.75} x$ continuous?

A5: Yes, the function $F(x)=\log_{0.75} x$ is continuous for all $x > 0$ .

Q6: Is the function $F(x)=\log_{0.75} x$ differentiable?

A6: Yes, the function $F(x)=\log_{0.75} x$ is differentiable for all $x > 0$ .

Q7: What is the domain of the function $F(x)=\log_{0.75} x$ ?

A7: The domain of the function $F(x)=\log_{0.75} x$ is all real numbers $x > 0$ .

Q8: What is the range of the function $F(x)=\log_{0.75} x$ ?

A8: The range of the function $F(x)=\log_{0.75} x$ is all real numbers $y$ such that $y < \log_{0.75} 1 = 0$ .

Q9: Is the function $F(x)=\log_{0.75} x$ one-to-one?

A9: Yes, the function $F(x)=\log_{0.75} x$ is one-to-one.

Q10: Can the function $F(x)=\log_{0.75} x$ be inverted?

A10: Yes, the function $F(x)=\log_{0.75} x$ can be inverted.

Conclusion

In conclusion, the function $F(x)=\log_{0.75} x$ is a decreasing function with a negative derivative for all $x > 0$ . It is continuous, differentiable, and one-to-one, and its domain and range are all real numbers $x > 0$ and $y < 0$ , respectively.

Final Answer

The final answer is:

A. Yes

The function $F(x)=\log_{0.75} x$ is indeed a decreasing function with a negative derivative for all $x > 0$ .