If $\log_{10000} x = -\frac{1}{4}$, then $x$ is equal to

Concept & Formula The fundamental definition of a logarithm connects it directly to exponents. If a logarithm is written as $\log_a b = c$, it can be converted to its exponential form: $$b = a^c$$ Step-by-Step Solution * **Given:** $\log_{10000} x = -\frac{1}{4}$ * **Conversion:** Convert the logarithmic equation into its equivalent exponential form. $$x = (10000)^{-\frac{1}{4}}$$ * **Prime Factorization:** Express the base $10000$ as a power of $10$. We know that $10000 = 10^4$. $$x = (10^4)^{-\frac{1}{4}}$$ * **Exponent Rules:** Apply the power of a power property, which states $(a^m)^n = a^{mn}$. $$x = 10^{4 \times -\frac{1}{4}}$$ * **Simplification:** The $4$ in the numerator and denominator cancel out. $$x = 10^{-1}$$ * **Final Value:** Use the negative exponent rule $a^{-n} = \frac{1}{a^n}$. $$x = \frac{1}{10}$$ Exam Strategy & Shortcut When dealing with large powers of $10$ and fractional exponents, always rewrite the large base as a power of $10$ first. Recognizing $10000$ as $10^4$ mentally allows you to immediately cancel the $4$ in the denominator of the exponent $-\frac{1}{4}$, jumping straight to $10^{-1}$ and skipping unnecessary written steps. Common Pitfall A frequent mistake is interpreting a negative exponent as making the entire number negative (e.g., thinking the answer is $-10$ or $-\frac{1}{10000}$). Remember that negative exponents dictate reciprocals, not negative values. Final Answer **Therefore, the correct answer is $\frac{1}{10}$.**