Find the number of digits in 8^57. (Given log10 2 = 0.3010)

Introduction / Context:We again use the digit-count identity: digits = floor(log10 N) + 1. Express 8^57 as a power of 2 to leverage the provided log10 2 value.

Given Data / Assumptions:

• N = 8^57 = (2^3)^57 = 2^171.
• log10 2 = 0.3010.

Concept / Approach:

• Compute log10(2^171) = 171 * 0.3010.
• Apply digits = floor(log10 N) + 1.

Step-by-Step Solution:

Verification / Alternative check:Since 10^51 < N < 10^52 and it is not an exact power of 10, the digit count must be 52.

Why Other Options Are Wrong:

• 50, 51, 53 result from rounding errors or dropping the +1.

Common Pitfalls:

• Using 0.3 instead of 0.3010 gives 51.3 and still yields 52 digits, but coarser approximations can mislead if rounding is mishandled.

Final Answer:52