I'm studying for state Mathcounts and found this problem solution in the 2014 state test:
- How many digits are in the integer representation of 2^30?
Looking at the powers of 2, we have 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768...
The number of digits in each can be written as: 1 1 1 2 2 2 3 3 3 etc.
Therefore, the number of digits needed to represent 230 is 10.
The numbers with 4 digits in the list provided obviously do not follow this rule: 1024 2048 4096 8192
However, the answer of 10 is correct, as 2^30 is 1073741824, which has 10 digits.
Note that calculators are not allowed on this test.
Is this process correct by some weird reasoning, or is the solution just lucky?
Also, my original solution included an experimental logarithm I performed. Log 2 of 10 yielded 3.32192809489, which I think means 2 to the power of 3.32193 is 10. Since each power of 10 includes 3.32192809489 powers of 2, I divided 30 by this ratio, yielding 9.03089986992. This would mean that 10 to the power of 9.03089986992 is approximately 2^30, which I verified with a calculator. Unfortunately this is not a viable way to solve without a calculator.
If the Mathcounts official solution is incorrect, is there a more practical solution to this problem without a calculator?
Breaking 2^30 into (2^10)^3 = 1024^3 makes this a simple estimation of the number of digits in 1024^3, which is 10.
I would still like to know if the answer provided by Mathcounts is a possible solution.