Number of digits of 2^64 using log10 2: Given log10(2) = 0.3010, find the number of decimal digits in 2^64.

Introduction / Context:The number of digits of a positive integer N equals ⌊log10 N⌋ + 1. With log10 2 given, we can compute log10(2^64) and apply this rule.

Given Data / Assumptions:

• log10 2 = 0.3010 (approximation).
• N = 2^64.

Concept / Approach:Use logarithm laws: log10(2^64) = 64·log10 2. Then number of digits = floor(value) + 1.

Step-by-Step Solution:log10(2^64) = 64 × 0.3010 = 19.264.Number of digits = ⌊19.264⌋ + 1 = 19 + 1 = 20.

Verification / Alternative check:2^10 ≈ 10^3 ⇒ 2^60 ≈ 10^18 and 2^4 = 16 ⇒ roughly 1.6 × 10^19 — a 20-digit number, consistent.

Why Other Options Are Wrong:19 would undercount; 21 would overcount compared to the logarithmic computation.

Common Pitfalls:Forgetting to add 1 after flooring, or rounding 19.264 up instead of taking the floor.

Final Answer:20