1. **State the problem:** Solve for $x$ in the equation $$\ln x + \log x = 5$$ where $\ln$ is the natural logarithm (base $e$) and $\log$ is the common logarithm (base 10). 2. **Recall the relationship between logarithms:** We know that $$\log x = \frac{\ln x}{\ln 10}$$ because changing the base of a logarithm uses the formula $$\log_a b = \frac{\log_c b}{\log_c a}$$. 3. **Rewrite the equation using natural logarithms:** $$\ln x + \frac{\ln x}{\ln 10} = 5$$ 4. **Factor out $\ln x$:** $$\ln x \left(1 + \frac{1}{\ln 10}\right) = 5$$ 5. **Simplify the factor:** $$1 + \frac{1}{\ln 10} = \frac{\ln 10 + 1}{\ln 10}$$ 6. **Rewrite the equation:** $$\ln x \cdot \frac{\ln 10 + 1}{\ln 10} = 5$$ 7. **Isolate $\ln x$:** $$\ln x = 5 \cdot \frac{\ln 10}{\ln 10 + 1}$$ 8. **Exponentiate both sides to solve for $x$:** $$x = e^{5 \cdot \frac{\ln 10}{\ln 10 + 1}}$$ 9. **Final answer:** $$\boxed{x = e^{5 \cdot \frac{\ln 10}{\ln 10 + 1}}}$$ This is the exact solution. You can approximate numerically if needed by substituting $\ln 10 \approx 2.302585$.