Question

Solve the system. Use any method you wish.

$\left\{\begin{array}{l}{\log }_x\left(2y\right)=3\\ {\log }_x\left(4y\right)=2\end{array}\right.$

Short answer

The solution of system of equations are $\left(\frac{1}{2},\frac{1}{16}\right)$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$\left\{\begin{array}{l}{\log }_x\left(2y\right)=3\\ {\log }_x\left(4y\right)=2\end{array}\right.$

Using logarithmic rules,

$$\begin{gathered} 2y=x^3......\left(i\right) \\ 4y=x^2......\left(ii\right) \end{gathered}$$

Rewrite equation (i),

$y=\frac{x^3}{2}$

Step 2. Substitute the value of y in equation (ii).

Substituting the value of y in equation (ii),

$$\begin{gathered} 4\left(\frac{x^3}{2}\right)=x^2 \\ 2x=1 \\ x=\frac{1}{2} \end{gathered}$$

Substituting the value of x in equation (ii),

$$\begin{gathered} 4y={\left(\frac{1}{2}\right)}^2 \\ y=\frac{1}{4}\times \frac{1}{4} \\ y=\frac{1}{16} \end{gathered}$$

Therefore, the solution sets are$\left(\frac{1}{2},\frac{1}{16}\right)$.