The value of $\log_{.01}(1000)$ is

Concept & Formula When both the base and the argument can be expressed as powers of the same number (e.g., $10$), use the base and argument exponent rule: $$ \log_{a^m} (a^n) = \frac{n}{m} $$ Step-by-Step Solution * **Given:** Evaluate $\log_{.01}(1000)$. * **Calculation:** Express both the base ($0.01$) and the argument ($1000$) as powers of $10$. * For the base: $0.01 = \frac{1}{100} = 10^{-2}$. * For the argument: $1000 = 10^3$. * Substitute these into the logarithm: $$ \log_{10^{-2}} (10^3) $$ * Apply the property $\log_{a^m} (b^n) = \frac{n}{m} \log_a b$: $$ \frac{3}{-2} \log_{10} 10 $$ * Since $\log_{10} 10 = 1$, the expression simplifies to: $$ -\frac{3}{2} $$ Exam Strategy & Shortcut Don't write out the formulas. Just look at the powers of $10$. The base $.01$ is $-2$ power. The argument $1000$ is $+3$ power. The answer is always (power of argument) / (power of base) $\rightarrow 3 / -2 = -3/2$. Common Pitfall Students often invert the fraction, arriving at $-2/3$ instead of $-3/2$. Always remember it is the argument's exponent divided by the base's exponent, not the other way around. Final Answer **Therefore, the correct answer is -\frac{3}{2}.**