If $\log_{10} 2 = 0.3010$, then $\log_{2} 10$ is equal to

### Concept & Formula This problem directly tests your knowledge of the Base-Switching property of logarithms. If you swap the base and the argument of a logarithm, the new logarithmic value is simply the mathematical reciprocal of the original value. Formula: $\log_a b = \frac{1}{\log_b a}$ ### Step-by-Step Solution Given the equation: $$ \log_{10} 2 = 0.3010 $$ Step 1: Apply the base-switching rule to find $\log_{2} 10$. $$ \log_{2} 10 = \frac{1}{\log_{10} 2} $$ Step 2: Substitute the known numerical value into the denominator. $$ \log_{2} 10 = \frac{1}{0.3010} $$ Step 3: Convert the decimal into a fraction to match the provided multiple-choice options. Since $0.3010$ can be written as $\frac{3010}{10000}$ or $\frac{301}{1000}$: $$ \log_{2} 10 = \frac{1}{\frac{301}{1000}} $$ Step 4: Simplify the complex fraction by flipping the denominator. $$ \log_{2} 10 = \frac{1000}{301} $$ ### Exam Strategy & Shortcut **Reciprocal Decimal Recognition:** When you see the base and argument flipped, immediately know you just need the reciprocal: $1 / 0.3010$. To calculate quickly without a calculator, ignore the decimal places temporarily. You know you are doing $1$ divided by roughly $3/10$, so the answer must be around $10/3$. Looking at the options, $\frac{1000}{301}$ is the exact fractional representation of this reciprocal. ### Common Pitfall Students often get confused by the zeros in the decimal and select an incorrectly scaled fraction like $\frac{100}{301}$ or mistakenly attempt to use subtraction rules. Keep your decimal places aligned when converting to a fraction: $0.301 = 301/1000$. ### Final Answer **Therefore, the correct answer is $\frac{1000}{301}$.**