Estimate the value of 1,000,000! in terms of a power of 10 and a power of e

Calculating the value of 1,000,000 factorial (1,000,000!) directly is impractical due to its enormous size. However, we can use Stirling’s approximation to estimate its logarithm and express the result in terms of powers of 10 and e.

Stirling’s approximation states that:

n! ≈ √(2πn) (n/e)ⁿ

For n = 1,000,000, we can rewrite this as:

1,000,000! ≈ √(2π(1,000,000)) (1,000,000/e)¹⁰⁰⁰⁰⁰⁰

Now, to find the logarithm base 10, we can compute:

log₁₀(1,000,000!) = log₁₀(√(2π(1,000,000))) + 1,000,000 log₁₀(1,000,000/e)

Calculating each part:

• √(2π(1,000,000)) ≈ 2,506.63
• log₁₀(2,506.63) ≈ 3.398

Next, we find:

• log₁₀(1,000,000) = 6 (since 1,000,000 = 10⁶)
• log₁₀(e) ≈ 0.434

Putting this all together:

log₁₀(1,000,000!) ≈ 3.398 + 1,000,000(6 – 0.434)

log₁₀(1,000,000!) ≈ 3.398 + 5,164,000 ≈ 5,164,003.398

Thus, we can express:

1,000,000! ≈ 10^{5,164,003.398}

Next, to estimate it in terms of a power of e, we can use:

log_e(1,000,000!) = log_e(√(2π(1,000,000))) + 1,000,000 log_e(1,000,000/e)

Using our previous results:

• log_e(2,506.63) ≈ 7.824
• 1,000,000(6 – 1) = 1,000,000 * 5 = 5,000,000

Therefore:

log_e(1,000,000!) ≈ 7.824 + 5,000,000 ≈ 5,000,007.824

This gives us:

1,000,000! ≈ e^{5,000,007.824}

In summary, the estimates for 1,000,000! can be expressed as:

• In terms of a power of 10: 1,000,000! ≈ 10^{5,164,003.398}
• In terms of a power of e: 1,000,000! ≈ e^{5,000,007.824}