If $\log 2 = 0.30103$, the number of digits in $4^{50}$ is

### Concept & Formula To find the number of digits in an exponential number, we use its base-10 logarithm. The number of digits is always one more than the characteristic (the integer part) of the logarithm. Before calculating, if the base of the exponent (in this case, $4$) can be simplified into a prime base we know (like $2$), we should apply exponent rules first. Formula: Number of digits = Characteristic $+ 1$ ### Step-by-Step Solution Given the value: $$ \log 2 = 0.30103 $$ Step 1: Simplify the target expression $4^{50}$ by converting the base $4$ into base $2$. $$ 4 = 2^2 $$ $$ 4^{50} = (2^2)^{50} $$ Step 2: Multiply the exponents. $$ (2^2)^{50} = 2^{100} $$ Step 3: Take the common logarithm of the simplified expression and apply the Power Rule. $$ \log (2^{100}) = 100 \times \log 2 $$ Step 4: Substitute the given value for $\log 2$. $$ 100 \times 0.30103 = 30.103 $$ Step 5: Determine the number of digits. The characteristic (integer part) is $30$. $$ \text{Number of digits} = 30 + 1 = 31 $$ ### Exam Strategy & Shortcut **Zero-Math Shift:** Recognizing $4^{50}$ as $2^{100}$ is the ultimate shortcut here. Multiplying any decimal by $100$ simply shifts the decimal point two places to the right. You instantly transform $0.30103$ into $30.103$. Grab the $30$, add $1$, and mark $31$ as your answer in under 5 seconds without touching a pen. ### Common Pitfall A less efficient method is to find $\log 4$ first ($2 \times 0.30103 = 0.60206$) and then multiply by $50$. While mathematically sound, multiplying a long decimal by $50$ introduces more room for simple arithmetic errors compared to simply shifting a decimal point with $100$. ### Final Answer **Therefore, the correct answer is 31.**