Let's solve the equation \(5^{2x+1} = 9^{x+1}\).
1. **Take the logarithm of both sides**:
\[\log(5^{2x+1}) = \log(9^{x+1})\]
2. **Apply the power rule of logarithms** (bringing the exponents down):
\[(2x+1) \cdot \log(5) = (x+1) \cdot \log(9)\]
3. **Expand and rearrange the equation**:
\[2x\log(5) + \log(5) = x\log(9) + \log(9)\]
4. **Group x terms on one side and constants on the other**:
\[2x\log(5) - x\log(9) = \log(9) - \log(5)\]
5. **Factor out x**:
\[x(2\log(5) - \log(9)) = \log(9) - \log(5)\]
6. **Solve for x**:
\[x = \frac{\log(9) - \log(5)}{2\log(5) - \log(9)}\]
This is the solution to the given equation. You may use a calculator to get the numerical approximation of this expression.
