If $\log_x y = 100$ and $\log_2 x = 10$, then the value of $y$ is

Concept & Formula This problem involves solving a system of sequential logarithmic equations. You must use the fundamental definition of logarithms twice: $$\log_a b = c \implies b = a^c$$ Additionally, you will apply the exponent rule for raising a power to a power: $(a^m)^n = a^{mn}$. Step-by-Step Solution * **Given:** Two equations: $\log_x y = 100$ and $\log_2 x = 10$. * **Solving the First Variable:** Start with the equation that has a known base to find $x$. $$\log_2 x = 10$$ * **Conversion:** Convert this to exponential form. $$x = 2^{10}$$ * **Substituting into Second Equation:** Now look at the other given equation: $\log_x y = 100$. Convert it to exponential form to isolate $y$. $$y = x^{100}$$ * **Combining Values:** Substitute the value of $x$ we found ($2^{10}$) into this new equation. $$y = (2^{10})^{100}$$ * **Simplification:** Multiply the exponents together using the power rule. $$y = 2^{10 \times 100}$$ $$y = 2^{1000}$$ Exam Strategy & Shortcut You can solve this rapidly using the change of base property or chaining. Recognize that $y = x^{100}$ and $x = 2^{10}$. Mental substitution $(2^{10})^{100}$ immediately yields $2^{1000}$. No intermediate calculations of $2^{10}$ ($1024$) are necessary since the options remain in exponential form. Common Pitfall A common mistake is adding the exponents instead of multiplying them when evaluating $(2^{10})^{100}$, leading to the incorrect answer $2^{110}$. Another trap is simply multiplying $100$ and $10$ as coefficients, completely misapplying logarithm rules. Always strictly follow the power of a power rule. Final Answer **Therefore, the correct answer is $2^{1000}$.**