1. **State the problem:** Solve the equations \( \log(x(x+21)^{10}) = 2 \) and \( \ln 2 - \ln (x+1) = 3 \). 2. **Recall logarithm properties:** - \( \log a = b \) means \( a = 10^b \) for common logarithm. - \( \ln a - \ln b = \ln \frac{a}{b} \). 3. **Solve the first equation:** \[ \log(x(x+21)^{10}) = 2 \implies x(x+21)^{10} = 10^2 = 100 \] This matches the given second equation. 4. **Solve the second equation:** \[ \ln 2 - \ln (x+1) = 3 \implies \ln \frac{2}{x+1} = 3 \implies \frac{2}{x+1} = e^3 \] 5. **Isolate \(x\):** \[ x+1 = \frac{2}{e^3} \implies x = \frac{2}{e^3} - 1 \] 6. **Summary:** - From the first equation, \( x(x+21)^{10} = 100 \). - From the second, \( x = \frac{2}{e^3} - 1 \). 7. **Check if \(x\) from the second satisfies the first:** Substitute \( x = \frac{2}{e^3} - 1 \) into \( x(x+21)^{10} \) and verify if it equals 100. **Final answers:** \[ x = \frac{2}{e^3} - 1 \quad \text{and} \quad x(x+21)^{10} = 100 \] These are the solutions to the given logarithmic equations.