If log 11=1.0414, prove that `10^11gt11^10`.

Answer

To prove that \(10^{11} > 11^{10}\) given that \(\log 11 = 1.0414\), we can follow these steps:

### Step 1: Define the Variables
Let:
- \(x = 10^{11}\)
- \(y = 11^{10}\)

### Step 2: Take Logarithm of Both Sides
We will take the logarithm (base 10) of both \(x\) and \(y\):
\[ \log x = \log(10^{11}) \quad \text{and} \quad \log y = \log(11^{10}) \]

### Step 3: Apply Logarithmic Properties
Using the property \(\log(a^b) = b \log a\), we can simplify:
\[ \log x = 11 \log 10 \quad \text{and} \quad \log y = 10 \log 11 \]

Since \(\log 10 = 1\):
\[ \log x = 11 \cdot 1 = 11 \]
And for \(y\):
\[ \log y = 10 \log 11 \]

### Step 4: Substitute the Given Value
We know from the problem that \(\log 11 = 1.0414\). Thus:
\[ \log y = 10 \cdot 1.0414 = 10.414 \]

### Step 5: Compare \(\log x\) and \(\log y\)
Now we compare \(\log x\) and \(\log y\):
\[ \log x = 11 \quad \text{and} \quad \log y = 10.414 \]

Since \(11 > 10.414\), we conclude:
\[ \log x > \log y \]

### Step 6: Exponentiate to Compare \(x\) and \(y\)
Since the logarithm is a monotonically increasing function, we can exponentiate both sides:
\[ x > y \]

### Step 7: Substitute Back the Values of \(x\) and \(y\)
Substituting back the definitions of \(x\) and \(y\):
\[ 10^{11} > 11^{10} \]

### Conclusion
Thus, we have proved that:
\[ 10^{11} > 11^{10} \]

---