1. **State the problem:** Solve the equation $3^x = x$ for $x$. 2. **Understand the problem:** We want to find the value(s) of $x$ such that the exponential function $3^x$ equals the linear function $x$. 3. **Analyze the functions:** The function $y=3^x$ grows exponentially and is always positive, while $y=x$ is a straight line. 4. **Check for obvious solutions:** At $x=0$, $3^0=1$ but $x=0$, so no equality. 5. **Check $x=1$:** $3^1=3$ and $x=1$, no equality. 6. **Check $x=3$:** $3^3=27$ and $x=3$, no equality. 7. **Check for negative values:** For $x=-1$, $3^{-1} = \frac{1}{3} \approx 0.333$ and $x=-1$, no equality. 8. **Graphical insight:** The exponential $3^x$ is always positive, but $x$ can be negative, so the only possible intersection points are where $x$ is positive. 9. **Use numerical methods:** Since no simple algebraic solution exists, approximate the solution numerically. 10. **Approximate solution:** Using trial, the solution is near $x \approx 0.7$ where $3^{0.7} \approx 0.7$. **Final answer:** The equation $3^x = x$ has one real solution approximately at $$x \approx 0.7$$.