Q.Given $\log 2 = 0.3010$, find the number of digits in $2^{50}$.

The number of digits in a whole number is one more than the characteristic of its logarithm, so we only need $\log(2^{50})$, never the number itself.

By the power law,

The characteristic of this logarithm is 15 (the integral part), and by the digit-count rule (§7) the number of digits in a whole number equals …

Cross-check by direct computation. $2^{50} = 1{,}125{,}899{,}906{,}842{,}624$. Counting the digits: 1-125-899-906-842-624 …

A common error is reporting the number of digits as equal to the characteristic (15) rather than characteristic + 1 (16). The digit count is always one more than the characteristic, because a characteristic of …