Answer
The simplified solution set for the system of equations is $\left\{ \left( {{10}^{4}},5 \right) \right\}$.
Work Step by Step
We use the substitution method to solve the provided system of equations,
$\left\{ \begin{align}
& \log {{x}^{2}}=y+3.....\text{(I)} \\
& \log x=y-1....\text{(II)}
\end{align} \right.$
Apply the logarithm rule ${{\log }_{m}}{{a}^{2}}=2{{\log }_{m}}a $ in equation (I),
$\begin{align}
& \log {{x}^{2}}=y+3 \\
& 2\log x=y+3 \\
& \log x=\frac{y+3}{2}
\end{align}$
Substitute the value of $\log x $ in equation (II),
$\begin{align}
& \log x=y-1 \\
& \frac{y+3}{2}=y-1 \\
\end{align}$
The above equation is in a single variable. To solve this equation, simplify as shown below:
$\begin{align}
& \frac{y+3}{2}=y-1 \\
& y+3=2\left( y-1 \right) \\
& y+3=2y-2 \\
& y=5
\end{align}$
The solution to this equation is $ y=5$.
Substitute the values of y in equation (II) to find values for x,
For, $ y=5$
$\begin{align}
& \log x=y-1 \\
& \log x=5-1 \\
& \log x=4 \\
\end{align}$
To solve this logarithmic equation, transform the equation to a power of 10 on both sides,
$\begin{align}
& \log x=4 \\
& {{10}^{\log x}}={{10}^{4}}
\end{align}$
Apply the logarithm rule ${{10}^{{{\log }_{{}}}a}}=a $ to simplify,
$\begin{align}
& {{10}^{\log x}}={{10}^{4}} \\
& x={{10}^{4}}
\end{align}$
Hence, for $ y=5$, we have $ x={{10}^{4}}$.
