1. **State the problem:** Solve the equation $$7^{x+2} - 7^{x+1} + 7^x = 86$$ for $x$. 2. **Rewrite the equation using properties of exponents:** Recall that $$7^{x+2} = 7^x \cdot 7^2 = 49 \cdot 7^x$$ and $$7^{x+1} = 7^x \cdot 7 = 7 \cdot 7^x$$. 3. **Substitute these into the equation:** $$49 \cdot 7^x - 7 \cdot 7^x + 7^x = 86$$ 4. **Factor out $7^x$:** $$7^x (49 - 7 + 1) = 86$$ 5. **Simplify inside the parentheses:** $$49 - 7 + 1 = 43$$ So the equation becomes: $$7^x \cdot 43 = 86$$ 6. **Divide both sides by 43:** $$\cancel{43} \cdot 7^x = \frac{86}{\cancel{43}}$$ $$7^x = 2$$ 7. **Solve for $x$ by taking the natural logarithm of both sides:** $$x = \frac{\ln 2}{\ln 7}$$ **Final answer:** $$S = \left\{ \frac{\ln 2}{\ln 7} \right\}$$