Using log10 3 = 0.47712, find the number of digits in the decimal representation of 3 raised to the power 33.

Introduction / Context:
This question tests the standard logarithm technique for finding how many decimal digits are in a very large power such as 3^33. Direct computation of 3^33 is not practical in an exam setting, so students must know the connection between the common logarithm log10 N and the number of digits in N. Such questions appear frequently in aptitude exams and engineering entrance tests, and they are a good check of both logarithm rules and basic understanding of place value in the decimal system.

Given Data / Assumptions:
- We are given log10 3 = 0.47712.
- We must find the number of digits in 3^33 when written in base 10.
- The number 3^33 is positive, so its common logarithm is defined.
- All logarithms are taken to base 10 unless otherwise specified.

Concept / Approach:
If N is a positive integer, then the number of decimal digits in N is given by digits = floor(log10 N) + 1. This follows from the fact that 10^(k) <= N < 10^(k+1) means N has k + 1 digits. For a power like 3^33 we first compute log10(3^33) using the power rule log10(a^n) = n * log10 a, and then apply the digit formula. The key idea is to avoid computing 3^33 directly and instead work entirely with logarithms.

Step-by-Step Solution:
Let N = 3^33. Compute log10 N using the power rule: log10 N = log10(3^33) = 33 * log10 3. Substitute the given value log10 3 = 0.47712. So log10 N = 33 * 0.47712. Multiply: 33 * 0.47712 = 15.74496. Therefore log10 N = 15.74496. The number of digits in N is floor(15.74496) + 1. floor(15.74496) = 15, so digits = 15 + 1 = 16.

Verification / Alternative check:
We know 10^15 has 16 digits and 10^16 has 17 digits. The value 3^33 has a common logarithm of approximately 15.74496, which lies between 15 and 16. That means 10^15 < 3^33 < 10^16, so 3^33 must have more digits than 10^15 but fewer digits than 10^16, giving exactly 16 digits. This confirms the result obtained from the floor and plus one formula.

Why Other Options Are Wrong:
14: This would correspond to log10(3^33) being between 13 and 14, which is much smaller than the computed value 15.74496.
15: This would require log10(3^33) to be between 14 and 15, which again does not match 15.74496.
17: This would require log10(3^33) to be at least 16, which is larger than 15.74496, so 17 digits is impossible.

Common Pitfalls:
A common mistake is to forget the plus one in the digit formula and answer floor(log10 N) instead of floor(log10 N) + 1. Another frequent error is to round 15.74496 to 16 and incorrectly use that rounded value directly as the number of digits. Students may also misapply logarithm rules, for example by writing log10(3^33) as (log10 3)^33, which is incorrect. Careful use of the power rule and the exact digit formula avoids these errors.

Final Answer:
The number of digits in 3^33 is 16.