Find the value of `log_(10)(10.1)` given that `log_(10) e=0.4343`.

Answer

To find the value of \( \log_{10}(10.1) \) given that \( \log_{10} e = 0.4343 \), we can use the concept of derivatives and the properties of logarithms. Here’s a step-by-step solution:

### Step 1: Define the function
Let \( f(x) = \log_{10}(x) \). We need to evaluate \( f(10.1) \).

### Step 2: Use the approximation formula
We can use the approximation:
\[ f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x \]
where \( x = 10 \) and \( \Delta x = 0.1 \).

### Step 3: Calculate \( f(10) \)
We know:
\[ f(10) = \log_{10}(10) = 1 \]

### Step 4: Find the derivative \( f'(x) \)
Using the change of base formula:
\[ f(x) = \frac{\log_e(x)}{\log_e(10)} \]
The derivative \( f'(x) \) is:
\[ f'(x) = \frac{1}{x \cdot \log_e(10)} \]

### Step 5: Evaluate \( f'(10) \)
Substituting \( x = 10 \):
\[ f'(10) = \frac{1}{10 \cdot \log_e(10)} \]

### Step 6: Use the known value of \( \log_{10} e \)
Since \( \log_{10} e = \frac{1}{\log_e(10)} \), we can express \( \log_e(10) \) as:
\[ \log_e(10) = \frac{1}{\log_{10} e} = \frac{1}{0.4343} \]

### Step 7: Substitute \( \log_e(10) \) in \( f'(10) \)
Thus:
\[ f'(10) = \frac{1}{10 \cdot \frac{1}{0.4343}} = \frac{0.4343}{10} = 0.04343 \]

### Step 8: Apply the approximation formula
Now, we can substitute back into the approximation:
\[ f(10.1) \approx f(10) + f'(10) \cdot 0.1 \]
\[ f(10.1) \approx 1 + 0.04343 \cdot 0.1 \]
\[ f(10.1) \approx 1 + 0.004343 = 1.004343 \]

### Final Answer
Thus, the value of \( \log_{10}(10.1) \) is approximately:
\[ \log_{10}(10.1) \approx 1.004343 \]