Which number is larger `(1.01)^(1000000) or 10,000` ?

Answer

To determine which number is larger between \( (1.01)^{1000000} \) and \( 10,000 \), we can use the properties of exponents and logarithms. Here’s a step-by-step solution:

### Step 1: Rewrite the comparison
We need to compare \( (1.01)^{1000000} \) and \( 10,000 \). To make the comparison easier, we can take the logarithm of both sides.

### Step 2: Take the logarithm
Taking the natural logarithm (or logarithm base 10) of both sides, we have:
\[ \log((1.01)^{1000000}) \quad \text{and} \quad \log(10,000) \]

### Step 3: Simplify using logarithm properties
Using the power rule of logarithms, we can simplify the left side:
\[ \log((1.01)^{1000000}) = 1000000 \cdot \log(1.01) \]
For the right side:
\[ \log(10,000) = \log(10^4) = 4 \]

### Step 4: Compare the two logarithmic expressions
Now we need to compare:
\[ 1000000 \cdot \log(1.01) \quad \text{and} \quad 4 \]

### Step 5: Calculate \( \log(1.01) \)
Using a calculator or logarithm table, we find:
\[ \log(1.01) \approx 0.004321 \]

### Step 6: Calculate \( 1000000 \cdot \log(1.01) \)
Now we can calculate:
\[ 1000000 \cdot \log(1.01) \approx 1000000 \cdot 0.004321 = 4321 \]

### Step 7: Compare the results
Now we compare:
\[ 4321 \quad \text{and} \quad 4 \]
Since \( 4321 > 4 \), we conclude that:
\[ 1000000 \cdot \log(1.01) > 4 \]

### Conclusion
Thus, we find that:
\[ (1.01)^{1000000} > 10,000 \]

### Final Answer
Therefore, \( (1.01)^{1000000} \) is larger than \( 10,000 \).

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