1. **State the problem:** Given that $3^x = 10$, find the value of $3^{x-3}$. 2. **Recall the exponent rule:** For any base $a$ and exponents $m$ and $n$, we have $$a^{m-n} = \frac{a^m}{a^n}.$$ This means $$3^{x-3} = \frac{3^x}{3^3}.$$ 3. **Substitute the known value:** Since $3^x = 10$, substitute this into the expression: $$3^{x-3} = \frac{10}{3^3}.$$ 4. **Calculate $3^3$:** $$3^3 = 3 \times 3 \times 3 = 27.$$ 5. **Simplify the expression:** $$3^{x-3} = \frac{10}{27}.$$ 6. **Final answer:** The value of $3^{x-3}$ is $$\boxed{\frac{10}{27}}.$$ This corresponds to option C.