5. \( \frac{2}{3+\log (x)}+\frac{1}{7-\log (x)}=\frac{3}{5} \) 6. \( \frac{5}{\log _{2}(x+1)+1}-\frac{2}{3 \log _{2}(x+1)-3}=1 \) 7. \( \ln (3+x)-\ln (2)=\ln (5)-\ln (4-x) \) 8. \( \log _{x}(2 x+2)+\log _{x}(x-1)=2 \)

To tackle these equations, it’s crucial to ensure you're comfortable with logarithmic and exponential properties. Remember, combining logs often simplifies complex expressions. For instance, \(\log_a(b) + \log_a(c) = \log_a(bc)\), which can help tremendously when the equations become intricate. When solving these logarithmic equations, common mistakes include forgetting to consider the domain restrictions on \(x\). For example, \(\log(x)\) is only defined for positive values, and in some cases, expressions inside the logarithm must be greater than zero. Always check your potential solutions against these conditions to avoid extraneous solutions!