Using differentials, find the approximate value of `log_(e)` (4.01), given that `log_(e)4 = 1.3863.`

Answer

To find the approximate value of \( \log_e(4.01) \) using differentials, we can follow these steps:

### Step-by-Step Solution

1. **Identify the function and known values:**
Let \( y = \log_e(x) \). We know that \( \log_e(4) = 1.3863 \).

2. **Define the change in \( x \):**
We want to find \( \log_e(4.01) \). We can express this as:
\[ x = 4 \quad \text{and} \quad \Delta x = 4.01 - 4 = 0.01 \]

3. **Differentiate the function:**
The derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{x} \]
At \( x = 4 \):
\[ \frac{dy}{dx} \bigg|_{x=4} = \frac{1}{4} \]

4. **Calculate \( dy \):**
Using the formula for differentials:
\[ dy = \frac{dy}{dx} \cdot dx \]
Substituting the values we have:
\[ dy = \frac{1}{4} \cdot 0.01 = 0.0025 \]

5. **Approximate \( \log_e(4.01) \):**
Now we can find \( \log_e(4.01) \) using the approximation:
\[ \log_e(4.01) \approx \log_e(4) + dy \]
Substituting the known values:
\[ \log_e(4.01) \approx 1.3863 + 0.0025 = 1.3888 \]

### Final Answer
The approximate value of \( \log_e(4.01) \) is \( 1.3888 \).

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