If log 2 = 0.30103 (base 10), then find the number of digits in 2⁵⁶.

Introduction / Context:
This question uses logarithms to find the number of digits in a large power of 2. Directly calculating 2⁵⁶ is not practical by hand, but the relationship between the logarithm of a number and its number of digits gives a simple and efficient method. This is a standard application of logarithms in number theory and competitive aptitude exams.

Given Data / Assumptions:

• log 2 (base 10) = 0.30103.
• We need to find the number of digits in 2⁵⁶.
• All logarithms have base 10.
• We use the formula: if N is a positive integer, the number of digits in N is equal to floor(log₁₀ N) + 1.

Concept / Approach:
First express 2⁵⁶ in terms of logs: log₁₀(2⁵⁶) = 56 log₁₀ 2. Once we compute this value, we apply the formula for the number of digits. The integer part of log₁₀ N tells us how many powers of 10 fit into N, and adding one gives the exact number of digits. This avoids any need for direct multiplication of large numbers.

Step-by-Step Solution:
Step 1: Use the given value log 2 = 0.30103 with base 10. Step 2: Compute log(2⁵⁶): log(2⁵⁶) = 56 log 2 = 56 × 0.30103. Step 3: Multiply 56 × 0.30103 = 16.85768. Step 4: According to the digit formula, if N = 2⁵⁶, the number of digits in N is floor(log N) + 1. Step 5: Here, floor(16.85768) = 16, so the number of digits = 16 + 1 = 17.

Verification / Alternative check:
We know approximate powers of 2: 2¹⁰ is about 1024 (4 digits), 2²⁰ is about 10⁶ (7 digits), 2³⁰ around 10⁹, and 2⁵⁰ around 10¹⁵. Therefore, 2⁵⁶ will be around 10¹⁶·⁸, which clearly has 17 digits. This matches the result obtained from the logarithm method. The approximation confirms that our calculation is reasonable and consistent with known powers.

Why Other Options Are Wrong:
If the number of digits were 19, 23, or 25, that would imply exponents near 18, 22, or 24 in base 10, which do not agree with log(2⁵⁶) being 16.85768. Option 21 also does not fit the digit formula. Only 17 matches floor(16.85768) + 1 and is therefore correct.

Common Pitfalls:
Some learners round log(2⁵⁶) to 17 and then add 1 to get 18, which is incorrect. It is crucial to use the floor of the logarithm, not a rounded value. Others may forget the plus one in the formula and answer 16 instead. Maintaining precision in the multiplication step and strictly applying floor(log N) + 1 is key to solving such problems accurately.

Final Answer:
The number of digits in 2⁵⁶ is 17.