Given log10 2 = 0.301 and log10 3 = 0.477, how many digits are there in the decimal representation of 108 raised to the power 10?

Introduction / Context:
This question asks for the number of decimal digits in a large power, 108^10, using approximate values of log10 2 and log10 3. It is a classic application of logarithms to digit counting and is commonly seen in competitive exams. The idea is to avoid direct computation of the power by using the relationship between the common logarithm and the number of digits of a positive integer.

Given Data / Assumptions:
- log10 2 = 0.301 (approximate value).
- log10 3 = 0.477 (approximate value).
- We must find the number of digits in 108^10.
- All logarithms are common logarithms in base 10.

Concept / Approach:
If N is a positive integer, the number of digits in N in base 10 is given by digits = floor(log10 N) + 1. Here N is 108^10. To compute log10 N, we first express 108 in terms of its prime factors 2 and 3, then use the power rule of logarithms. Since 108 = 2^2 * 3^3, we can find log10 108 in terms of log10 2 and log10 3. Then log10 108^10 = 10 * log10 108. Finally, applying the digit formula gives the required number of digits.

Step-by-Step Solution:
First factor 108: 108 = 2^2 * 3^3. Compute log10 108 using the product rule and the power rule. log10 108 = log10 (2^2 * 3^3). Apply the product rule: log10 (2^2 * 3^3) = log10 (2^2) + log10 (3^3). Use the power rule: log10 (2^2) = 2 log10 2 and log10 (3^3) = 3 log10 3. Substitute the given values: log10 108 = 2(0.301) + 3(0.477). Compute: 2(0.301) = 0.602 and 3(0.477) = 1.431. So log10 108 is approximately 0.602 + 1.431 = 2.033. Now compute log10 (108^10) = 10 * log10 108. Thus log10 (108^10) is approximately 10 * 2.033 = 20.33. The number of digits in 108^10 is floor(20.33) + 1 = 20 + 1 = 21.

Verification / Alternative check:
We know that 10^20 has 21 digits and 10^21 has 22 digits. The value 108^10 has log10 value about 20.33, which lies between 20 and 21. This implies that 10^20 < 108^10 < 10^21, so 108^10 must have more digits than 10^20 but fewer digits than 10^21. Therefore it has exactly 21 digits. This confirms the result obtained from the floor and plus one formula and shows that small variations in the decimal approximations of log10 2 and log10 3 will not change the digit count.

Why Other Options Are Wrong:
19: This would require log10 (108^10) to be between 18 and 19, which is far from the computed value of about 20.33.
20: This would require log10 (108^10) to be between 19 and 20, which again does not match 20.33.
22: This would require log10 (108^10) to be at least 21, which is larger than the computed 20.33, so it is impossible.

Common Pitfalls:
Students sometimes round intermediate values too aggressively, for example rounding log10 108 to 2 instead of 2.033, which may still give the correct digit count but can be risky in closer cases. Another error is to forget the plus one in the digit formula and answer floor(log10 N) instead of floor(log10 N) + 1. Some also misfactor 108, for example writing 108 = 2^3 * 3^2, which would lead to an incorrect log10 108 and hence an incorrect digit count. Carefully factoring 108 and applying the log rules step by step prevents these mistakes.

Final Answer:
The number of digits in 108^10 is 21.