If log 2 = 0.301 and log 3 = 0.4771, find the number of digits in 4812.

Step-by-Step Solution

To find the number of digits in a large power like 48 to the power of 12, we can use logarithms. The rule is that the number of digits in a positive integer N is given by the formula: floor(log10(N)) + 1.

1. Set up the expression: Let N = 48 to the power of 12. We need to find log10(N) = log10(48 to the power of 12). Using log rules, this is 12 * log10(48).

2. Break down the logarithm: We can express 48 as 16 * 3, or (2 to the power of 4) * 3. So, log10(48) = log10(2 to the power of 4 * 3). Using the property log(a * b) = log(a) + log(b), we get: log10(48) = log10(2 to the power of 4) + log10(3) log10(48) = 4 * log10(2) + log10(3).

3. Substitute the given values: Given log10(2) = 0.301 and log10(3) = 0.4771: log10(48) = 4 * (0.301) + 0.4771 log10(48) = 1.204 + 0.4771 = 1.6811.

4. Calculate log10(N): log10(N) = 12 * 1.6811 = 20.1732.

5. Find the number of digits: The number of digits is floor(20.1732) + 1 = 20 + 1 = 21.