AMC10 2016 10A, Problem 2

For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$?

Solution: We can rewrite $10^{x}\cdot 100^{2x}=1000^{5}$ as $10^{5x}=10^{15}$: \[\begin{split} 10^x\cdot100^{2x} & =10^x\cdot(10^2)^{2x} \\ 10^x\cdot10^{4x} & =(10^3)^5 \\ 10^{5x} & =10^{15} \end{split}\] Since the bases are equal, we can set the exponents equal, giving us $5x=15$. Solving the equation gives us $x = \fbox{3}.$

Answer: 3