Solke simultaneous Find the value of \( m \) and \( n \) \( \log m+2 \log n=6 \) \( \log m-\log n=4 \)

Solve the system of equations \( \log(m)+2\log(n)=6;\log(m)-\log(n)=4 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}\log_{10}{\left(m\right)}+2\log_{10}{\left(n\right)}=6\\\log_{10}{\left(m\right)}-\log_{10}{\left(n\right)}=4\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}m=\frac{10^{6}}{n^{2}}\\\log_{10}{\left(m\right)}-\log_{10}{\left(n\right)}=4\end{array}\right.\) - step2: Substitute the value of \(m:\) \(\log_{10}{\left(\frac{10^{6}}{n^{2}}\right)}-\log_{10}{\left(n\right)}=4\) - step3: Subtract the terms: \(\log_{10}{\left(\frac{10^{6}}{n^{3}}\right)}=4\) - step4: Convert the logarithm into exponential form: \(\frac{10^{6}}{n^{3}}=10^{4}\) - step5: Evaluate the power: \(\frac{10^{6}}{n^{3}}=10000\) - step6: Cross multiply: \(10^{6}=n^{3}\times 10000\) - step7: Simplify the equation: \(10^{6}=10000n^{3}\) - step8: Rewrite the expression: \(10000\times 100=10000n^{3}\) - step9: Evaluate: \(100=n^{3}\) - step10: Swap the sides: \(n^{3}=100\) - step11: Simplify the equation: \(\sqrt[3]{n^{3}}=\sqrt[3]{100}\) - step12: Calculate: \(n=\sqrt[3]{100}\) - step13: Substitute the value of \(n:\) \(m=\frac{10^{6}}{\left(\sqrt[3]{100}\right)^{2}}\) - step14: Simplify: \(m=10^{4}\sqrt[3]{100}\) - step15: Calculate: \(m=10000\sqrt[3]{100}\) - step16: Calculate: \(\left\{ \begin{array}{l}m=10000\sqrt[3]{100}\\n=\sqrt[3]{100}\end{array}\right.\) - step17: Check the solution: \(\left\{ \begin{array}{l}m=10000\sqrt[3]{100}\\n=\sqrt[3]{100}\end{array}\right.\) - step18: Rewrite: \(\left(m,n\right) = \left(10000\sqrt[3]{100},\sqrt[3]{100}\right)\) The solution to the system of equations is \( m = 10000\sqrt[3]{100} \) and \( n = \sqrt[3]{100} \).