**Find \[\dfrac{d}{dx}[-8\log(x)]\].**

Choose 1 answer:

Choose 1 answer:

(Choice A)   

\[-\dfrac{8}{\ln(10)x}\]

A

\[-\dfrac{8}{\ln(10)x}\]

(Choice B)   

\[-\dfrac{8\ln(10)}{\ln(x)}\]

B

\[-\dfrac{8\ln(10)}{\ln(x)}\]

(Choice C)   

\[-\dfrac{8}{\log(x)}\]

C

\[-\dfrac{8}{\log(x)}\]

(Choice D)   

\[-\dfrac{8}{x}\]

D

\[-\dfrac{8}{x}\]




To solve \(\frac{d}{dx}\left[-8\log(x)\right]\), we need to differentiate the function using the rules for logarithms.

### Step-by-Step Solution:

1. **Recall the derivative of the logarithmic function:**
   \[
   \frac{d}{dx} \left[\log(x)\right] = \frac{1}{x\ln(10)}
   \]
   where \(\log(x)\) is the logarithm base 10.

2. **Apply the derivative rule:**
   The function to differentiate is \(-8\log(x)\). The constant factor \(-8\) can be factored out:
   \[
   \frac{d}{dx}\left[-8\log(x)\right] = -8 \cdot \frac{d}{dx}\left[\log(x)\right]
   \]

   Using the derivative of \(\log(x)\) we found earlier:
   \[
   \frac{d}{dx}\left[-8\log(x)\right] = -8 \cdot \frac{1}{x\ln(10)} = -\frac{8}{x\ln(10)}
   \]

The correct answer is:
\[
\textbf{(A) } -\dfrac{8}{\ln(10)x}
\]

Would you like more details on this solution, or do you have any questions?

Here are some related questions you might find interesting:

1. How do you differentiate logarithmic functions with different bases?
2. What is the difference between \(\ln(x)\) and \(\log(x)\)?
3. How does the chain rule apply to logarithmic functions?
4. What is the derivative of \(\log_b(x)\) for an arbitrary base \(b\)?
5. How does the constant factor affect the differentiation process?

**Tip:** Always remember that \(\log(x)\) typically refers to the logarithm with base 10, and \(\ln(x)\) refers to the natural logarithm (base \(e\)).

Learn how to differentiate -8log(x) step-by-step with the correct formula application. Explore the derivative of logarithmic functions and the chain rule. Suitable for advanced high school or college students.

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