Is the logarithmic function , where  an increasing or decreasing function, or neither? Explain using .

To determine whether the logarithmic function is increasing, decreasing, or neither, we can analyze its derivative. The logarithmic function is typically expressed as $f(x) = \log_a(x)$, where $a$ is the base of the logarithm. The most common bases used are $e$ (natural logarithm) and $10$ (common logarithm).

Step 1: Break Down the Problem

We need to find the derivative of the logarithmic function and analyze its sign to conclude whether the function is increasing or decreasing.

Step 2: Relevant Concepts

The properties of derivatives are essential here. Specifically:

• If $f'(x) > 0$, the function $f(x)$ is increasing.
• If $f'(x) < 0$, the function $f(x)$ is decreasing.
• If $f'(x) = 0$, the function is neither increasing nor decreasing.

Step 3: Analysis and Detail

Calculate the derivative of the logarithmic function:

For $f(x) = \log_a(x)$, the derivative is given by: $f'(x) = \frac{1}{x \ln(a)}$ where $\ln(a)$ is the natural logarithm of the base $a$.

• $f'(x) > 0$ when $x > 0$ because $x$ is always positive for the logarithm to be defined, and $\ln(a)$ is positive for $a > 1$.
• Therefore, since $\ln(a) > 0$, $f'(x)$ is always positive for $x > 0$.

Step 4: Verify and Summarize

As we verified, $f'(x) = \frac{1}{x \ln(a)}$ is positive for $x > 0$ and $a > 1$. Thus, the logarithmic function is increasing on its domain $(0, \infty)$.

Final Answer

The logarithmic function $f(x) = \log_a(x)$ is an increasing function for $a > 1$.