1. **State the problem:** We need to find the value of $x$ in the equation $103^x = 67$ where the base is 10. 2. **Formula and rules:** To solve for $x$ when the variable is in the exponent, we use logarithms. The key formula is: $$x = \log_{103}(67)$$ Since calculators typically use base 10 or base $e$ logarithms, we use the change of base formula: $$x = \frac{\log_{10}(67)}{\log_{10}(103)}$$ 3. **Calculate intermediate values:** Calculate $\log_{10}(67)$ and $\log_{10}(103)$: $$\log_{10}(67) \approx 1.8261$$ $$\log_{10}(103) \approx 2.0128$$ 4. **Evaluate $x$:** $$x = \frac{1.8261}{2.0128} \approx 0.9075$$ 5. **Interpretation:** The value of $x$ is approximately $0.9075$. This means $103$ raised to the power $0.9075$ equals $67$. **Final answer:** $$x \approx 0.9075$$