Question

Differentiate \( \log_a x \) with respect to \( a^x \)

• A. \( \frac{1}{x a^x} \)
• B. \( \frac{1}{x a^x (\log a)^2} \)
• C. \( \frac{\alpha}{x (\log a)^2} \)
• D. \( \frac{a^x}{x} \)

Hint:

When differentiating logarithmic functions with non-natural bases, express them in terms of the natural logarithm and apply the chain and quotient rules as needed.

Answer 1

The Correct Option is B

Step 1: Use the formula for \( \log_a x \).
We know that the logarithmic function \( \log_a x \) can be written in terms of the natural logarithm as:
\[ \log_a x = \frac{\ln x}{\ln a} \]

Step 2: Apply the chain rule.
We need to differentiate \( \log_a x \) with respect to \( a^x \). Using the chain rule:
\[ \frac{d}{dx} \log_a x = \frac{d}{dx} \left( \frac{\ln x}{\ln a} \right) \]
Since \( \ln a \) is constant with respect to \( x \), we get: \[ \frac{1}{\ln a} \cdot \frac{d}{dx} (\ln x) \]

Step 3: Differentiate \( \ln x \).
The derivative of \( \ln x \) with respect to \( x \) is: \[ \frac{1}{x} \]
Thus, \[ \frac{d}{dx} \log_a x = \frac{1}{x \ln a} \]

Step 4: Differentiate \( a^x \) with respect to \( x \).
Next, we differentiate \( a^x \) with respect to \( x \). Using the derivative formula for exponential functions: \[ \frac{d}{dx} a^x = a^x \ln a \]

Step 5: Apply the quotient rule.
We now apply the quotient rule to differentiate \( \log_a x \) with respect to \( a^x \). The quotient rule states:
\[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x) \frac{d}{dx} f(x) - f(x) \frac{d}{dx} g(x)}{(g(x))^2} \]
Substituting \( f(x) = \ln x \) and \( g(x) = a^x \), we get:
\[ \frac{d}{dx} \log_a x = \frac{a^x \cdot \frac{1}{x} - \ln x \cdot a^x \ln a}{(a^x)^2} \]

Step 6: Simplify the expression.
Simplifying the result:
\[ \frac{d}{dx} \log_a x = \frac{1}{x a^x (\log a)^2} \]
Thus, the correct answer is option (B).