Which number is larger `(1.1)^(100000) or 10,000` ?

Answer

To determine which number is larger between \( (1.1)^{100000} \) and \( 10,000 \), we can follow these steps:

### Step 1: Rewrite the problem
We need to compare \( (1.1)^{100000} \) and \( 10,000 \).

### Step 2: Take the logarithm
Taking the logarithm of both sides can help us compare the two numbers more easily. We can use the natural logarithm (ln) or logarithm base 10 (log). Here, we will use logarithm base 10 for simplicity.

Let's denote:
- \( A = (1.1)^{100000} \)
- \( B = 10,000 \)

We want to compare \( \log_{10}(A) \) and \( \log_{10}(B) \).

### Step 3: Calculate \( \log_{10}(A) \)
Using the properties of logarithms:
\[ \log_{10}(A) = \log_{10}((1.1)^{100000}) = 100000 \cdot \log_{10}(1.1) \]

### Step 4: Calculate \( \log_{10}(B) \)
Now, calculate \( \log_{10}(B) \):
\[ \log_{10}(B) = \log_{10}(10,000) = \log_{10}(10^4) = 4 \]

### Step 5: Compare the two logarithmic values
Now we need to compare \( 100000 \cdot \log_{10}(1.1) \) and \( 4 \).

### Step 6: Estimate \( \log_{10}(1.1) \)
Using a calculator or logarithm tables, we find:
\[ \log_{10}(1.1) \approx 0.0414 \]

### Step 7: Calculate \( 100000 \cdot \log_{10}(1.1) \)
Now we can calculate:
\[ 100000 \cdot \log_{10}(1.1) \approx 100000 \cdot 0.0414 = 4140 \]

### Step 8: Compare the results
Now we compare:
\[ 4140 \text{ (which is } 100000 \cdot \log_{10}(1.1) \text{) } > 4 \text{ (which is } \log_{10}(10,000) \text{)} \]

### Conclusion
Since \( 100000 \cdot \log_{10}(1.1) > 4 \), we conclude that:
\[ (1.1)^{100000} > 10,000 \]

Thus, \( (1.1)^{100000} \) is larger than \( 10,000 \).

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