Solve the systems in Exercises 79–80.
$\{\begin{array}{c}{\log }_y\text{ x=3}\\ {\log }_y\text{ (4x)=5}\end{array}$

Verified step by step guidance

1. Start by writing down the given system of logarithmic equations: $\log y_{}x=\; 3$ and $\log y_{}\left(4x\right)=\; 5$.
2. Recall the definition of logarithms: $\log a_{}b=\; c$ means $a^c\; =\; b$. Use this to rewrite each logarithmic equation in exponential form.
3. Rewrite the first equation: $y^3\; =\; x$. Rewrite the second equation: $y^5\; =\; 4x$.
4. Substitute the expression for $x$ from the first equation into the second equation: replace $x$ in $y^5\; =\; 4x$ with $y^3$, resulting in $y^5\; =\; 4y^3$.
5. Solve the resulting equation for $y$ by dividing both sides by $y^3$ (assuming $y\; eq\; 0$), which gives $y^\{5-3\}\; =\; 4$ or $y^2\; =\; 4$. Then find the possible values of $y$.

Key Concepts

Logarithmic Functions and Their Properties

A logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding properties like log_b(xy) = log_b(x) + log_b(y) and log_b(x^k) = k log_b(x) is essential for manipulating and solving logarithmic equations.

Change of Base and Variable Identification

In equations involving logs with unknown bases or arguments, recognizing how to express variables and rewrite equations using properties or substitutions helps isolate variables. Here, identifying y as the base and expressing x in terms of y is key to solving the system.

Solving Systems of Equations

A system of equations involves finding values that satisfy all equations simultaneously. Techniques include substitution or elimination. For logarithmic systems, converting logs to exponential form often simplifies the process and reveals relationships between variables.