What is the solution to $$ \log _{2}\left(2 x^{3}-8 ight)-2 \log _{2} x=\log _{2} x ? $$ $$ \begin{array}{c}x=1 \ x=2 \ 0 \ x=3 \ x=4\end{array} $$

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What is the solution to $$ \log _{2}\left(2 x^{3}-8ight)-2 \log _{2} x=\log _{2} x ? $$ $$ \begin{array}{c}x=1 \ x=2 \ 0 \ x=3 \ x=4\end{array} $$

UpStudy AI · Accepted Answer

Solve the equation $$ \log _{2}\left(2 x^{3}-8ight)-2 \log _{2} x=\log _{2} x $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{2}{\left(2x^{3}-8ight)}-2\log_{2}{\left(xight)}=\log_{2}{\left(xight)}$$
- step1: Find the domain:
  $$\log_{2}{\left(2x^{3}-8ight)}-2\log_{2}{\left(xight)}=\log_{2}{\left(xight)},x>\sqrt[3]{4}$$
- step2: Add the terms:
  $$\log_{2}{\left(\frac{2x^{3}-8}{x^{2}}ight)}=\log_{2}{\left(xight)}$$
- step3: Set the arguments equal:
  $$\frac{2x^{3}-8}{x^{2}}=x$$
- step4: Cross multiply:
  $$2x^{3}-8=x^{2}	imes x$$
- step5: Simplify the equation:
  $$2x^{3}-8=x^{3}$$
- step6: Move the expression to the left side:
  $$2x^{3}-8-x^{3}=0$$
- step7: Subtract the terms:
  $$x^{3}-8=0$$
- step8: Move the constant to the right side:
  $$x^{3}=0+8$$
- step9: Remove 0:
  $$x^{3}=8$$
- step10: Simplify the equation:
  $$\sqrt[3]{x^{3}}=\sqrt[3]{8}$$
- step11: Calculate:
  $$x=\sqrt[3]{8}$$
- step12: Evaluate:
  $$x=2$$
- step13: Check if the solution is in the defined range:
  $$x=2,x>\sqrt[3]{4}$$
- step14: Find the intersection:
  $$x=2$$
The solution to the equation $$ \log _{2}\left(2 x^{3}-8ight)-2 \log _{2} x=\log _{2} x $$ is $$ x=2 $$.
The solution is $$ x=2 $$.

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