The value of `lim_(x to a)(log(x-a))/log(e^(x) - e^(a))` is:

Options

A. 1
B. `-1`
C. 0
D. none of these

Answer

To solve the limit \( \lim_{x \to a} \frac{\log(x - a)}{\log(e^x - e^a)} \), we will follow these steps:

### Step 1: Direct Substitution
First, we will substitute \( x = a \) directly into the limit:
\[ \lim_{x \to a} \frac{\log(x - a)}{\log(e^x - e^a)} = \frac{\log(a - a)}{\log(e^a - e^a)} = \frac{\log(0)}{\log(0)} \]
This results in the indeterminate form \( \frac{-\infty}{-\infty} \).

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that if \( \lim_{x \to c} \frac{f(x)}{g(x)} \) results in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then:
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]
Here, \( f(x) = \log(x - a) \) and \( g(x) = \log(e^x - e^a) \).

### Step 3: Differentiate the Numerator and Denominator
Now we differentiate \( f(x) \) and \( g(x) \):
- The derivative of the numerator:
\[ f'(x) = \frac{1}{x - a} \]
- The derivative of the denominator:
\[ g'(x) = \frac{1}{e^x - e^a} \cdot e^x = \frac{e^x}{e^x - e^a} \]

### Step 4: Rewrite the Limit
Now we can rewrite the limit using the derivatives:
\[ \lim_{x \to a} \frac{f'(x)}{g'(x)} = \lim_{x \to a} \frac{\frac{1}{x - a}}{\frac{e^x}{e^x - e^a}} = \lim_{x \to a} \frac{(e^x - e^a)}{(x - a) e^x} \]

### Step 5: Substitute Again
Substituting \( x = a \) again gives us:
\[ = \frac{(e^a - e^a)}{(a - a)e^a} = \frac{0}{0} \]
This is still an indeterminate form, so we apply L'Hôpital's Rule again.

### Step 6: Differentiate Again
We differentiate the numerator and denominator again:
- The derivative of the numerator \( e^x - e^a \) is \( e^x \).
- The derivative of the denominator \( (x - a)e^x \) using the product rule is:
\[ e^x + (x - a)e^x = e^x(1 + (x - a)) \]

### Step 7: Rewrite the Limit Again
Now we rewrite the limit:
\[ \lim_{x \to a} \frac{e^x}{e^x(1 + (x - a))} = \lim_{x \to a} \frac{1}{1 + (x - a)} \]

### Step 8: Final Substitution
Substituting \( x = a \):
\[ = \frac{1}{1 + (a - a)} = \frac{1}{1} = 1 \]

### Conclusion
Thus, the value of the limit is:
\[ \lim_{x \to a} \frac{\log(x - a)}{\log(e^x - e^a)} = 1 \]