Find the number of digits in 8^10. (Given log10 2 = 0.3010)

Introduction / Context:Digit count problems rely on logarithms. If N > 0, then the number of decimal digits is floor(log10 N) + 1. We are given log10 2, which lets us evaluate powers of 2 efficiently.

Given Data / Assumptions:

• N = 8^10.
• 8 = 2^3, so 8^10 = (2^3)^10 = 2^30.
• log10 2 = 0.3010 (approximation sufficient for digit counting).

Concept / Approach:

• Compute log10(2^30) = 30 * log10 2.
• Use the digit-count formula: digits = floor(log10 N) + 1.

Step-by-Step Solution:

Verification / Alternative check:Since 10^9 = 1,000,000,000 and 10^10 = 10,000,000,000, and 8^10 ≈ 1.07 × 10^9, it lies between 10^9 and 10^10, confirming 10 digits.

Why Other Options Are Wrong:

• 19, 17, 20 misapply the digit formula or misuse log10 2.
• “None of these” is incorrect because 10 is exact.

Common Pitfalls:

• Forgetting to add 1 after flooring log10 N.

Final Answer:10