# 2016 AMC 10A Problems/Problem 2

## Problem

For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

## Solution 1

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 $x = \boxed{\textbf{(C)}\;3}.$

## Solution 2

We can rewrite this expression as $\log(10^x \cdot 100^{2x})=\log(1000^5)$ , which can be simplified to $\log(10^{x}\cdot10^{4x})=5\log(1000)$, and that can be further simplified to $\log(10^{5x})=5\log(10^3)$ . This leads to $5x=15$. Solving this linear equation yields $x = \boxed{\textbf{(C)}\;3}.$

## Video Solution

~IceMatrix

~savannahsolver

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 