If $\log 2 = 0.30103$, the number of digits in $2^{64}$ is

### Concept & Formula The number of digits in an integer $N$ can be directly determined using its common logarithm (base 10). When you calculate $\log_{10}(N)$, the whole number part of the result (the characteristic) determines the digits. Formula: Number of digits = Characteristic $+ 1$. ### Step-by-Step Solution Given the value: $$ \log 2 = 0.30103 $$ Step 1: Take the common logarithm of the target number, $2^{64}$. $$ \text{Let } x = 2^{64} $$ $$ \log x = \log (2^{64}) $$ Step 2: Apply the Power Rule of logarithms to bring the exponent to the front. $$ \log (2^{64}) = 64 \times \log 2 $$ Step 3: Substitute the known value of $\log 2$ and multiply. $$ 64 \times 0.30103 = 19.26592 $$ Step 4: Identify the characteristic and calculate the number of digits. The integral part (characteristic) of $19.26592$ is $19$. $$ \text{Number of digits} = 19 + 1 = 20 $$ ### Exam Strategy & Shortcut **Approximation Method:** You don't need the exact decimal value to find the characteristic. Round $\log 2$ to $0.3$ for a quick mental check. $$ 64 \times 0.3 = 19.2 $$ The whole number is $19$. Add $1$ to get $20$. Even if the actual value was slightly higher, it would not cross into the $20.x$ range, making this approximation incredibly safe and fast for competitive exams. ### Common Pitfall Students often forget the final step of adding $1$ to the characteristic. They correctly calculate $19.26$, identify $19$, and mistakenly select 19 as the final answer. Always remember: Digits = Characteristic $+ 1$. ### Final Answer **Therefore, the correct answer is 20.**