Solve the systems in Exercises 79–80.
$\{\begin{array}{c}\text{log }{x}^2=y+3\\ \text{log }{x}^{}=y-1\end{array}$

Verified step by step guidance

1. Rewrite the given system of equations for clarity: $\log x^2=\; y\; +\; 3$ and $\log x=\; y\; -\; 1$.
2. Recall the logarithm property: $\log x^2=\; 2\log x$. Use this to rewrite the first equation as $2\log x\; =\; y\; +\; 3$.
3. Substitute $\log x$ from the second equation into the first. Since $\log x\; =\; y\; -\; 1$, replace $\log x$ in the first equation with $y\; -\; 1$ to get $2(y\; -\; 1)\; =\; y\; +\; 3$.
4. Solve the resulting linear equation for $y$: expand and simplify $2y\; -\; 2\; =\; y\; +\; 3$, then isolate $y$.
5. Once you find $y$, substitute it back into $\log x\; =\; y\; -\; 1$ to find $\log x$, and then solve for $x$ by rewriting the logarithmic equation in exponential form.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the power rule (log a^b = b log a), is essential for manipulating and simplifying logarithmic expressions. This allows rewriting terms like log x^2 as 2 log x, facilitating easier comparison and solving of equations.

Solving Systems of Equations

Solving systems of equations involves finding values for variables that satisfy all given equations simultaneously. Techniques include substitution and elimination, which help reduce the system to a single-variable equation for easier solution.

Relationship Between Logarithmic and Linear Expressions

Recognizing how logarithmic expressions relate to linear equations is crucial. For example, expressing log x in terms of y allows converting the system into linear form, making it easier to solve using algebraic methods.