24) \( 9 \cdot 5^{2 v}=30 \)

Solve the equation \( 9 \cdot 5^{2v}=30 \). Solve the equation by following steps: - step0: Solve for \(v\): \(9\times 5^{2v}=30\) - step1: Divide both sides: \(\frac{9\times 5^{2v}}{9}=\frac{30}{9}\) - step2: Divide the numbers: \(5^{2v}=\frac{10}{3}\) - step3: Take the logarithm of both sides: \(\log_{5}{\left(5^{2v}\right)}=\log_{5}{\left(\frac{10}{3}\right)}\) - step4: Evaluate the logarithm: \(2v=\log_{5}{\left(\frac{10}{3}\right)}\) - step5: Divide both sides: \(\frac{2v}{2}=\frac{\log_{5}{\left(\frac{10}{3}\right)}}{2}\) - step6: Divide the numbers: \(v=\frac{\log_{5}{\left(\frac{10}{3}\right)}}{2}\) - step7: Simplify: \(v=\frac{1+\log_{5}{\left(2\right)}-\log_{5}{\left(3\right)}}{2}\) The solution to the equation \(9 \cdot 5^{2v} = 30\) is \(v = \frac{1 + \log_{5}(2) - \log_{5}(3)}{2}\).