What is the true solution to the logarithmic equation below? $$ \begin{array}{l}\log _{4}\left[\log _{4}(2 x) ight]=1 \ \begin{array}{l}x=2 \ x=8 \ x=64 \ x=128\end{array}\end{array} \begin{array}{l}\qquad\end{array} $$

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What is the true solution to the logarithmic equation below? $$ \begin{array}{l}\log _{4}\left[\log _{4}(2 x)ight]=1 \ \begin{array}{l}x=2 \ x=8 \ x=64 \ x=128\end{array}\end{array} \begin{array}{l}\qquad\end{array} $$

UpStudy AI · Accepted Answer

Solve the equation $$ \log_{4}(\log_{4}(2x))=1 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{4}{\left(\log_{4}{\left(2x\right)}\right)}=1$$
- step1: Find the domain:
  $$\log_{4}{\left(\log_{4}{\left(2x\right)}\right)}=1,x>\frac{1}{2}$$
- step2: Convert the logarithm into exponential form:
  $$\log_{4}{\left(2x\right)}=4^{1}$$
- step3: Evaluate the power:
  $$\log_{4}{\left(2x\right)}=4$$
- step4: Convert the logarithm into exponential form:
  $$2x=4^{4}$$
- step5: Evaluate the power:
  $$2x=256$$
- step6: Divide both sides:
  $$\frac{2x}{2}=\frac{256}{2}$$
- step7: Divide the numbers:
  $$x=128$$
- step8: Check if the solution is in the defined range:
  $$x=128,x>\frac{1}{2}$$
- step9: Find the intersection:
  $$x=128$$
The true solution to the logarithmic equation $$ \log_{4}(\log_{4}(2x))=1 $$ is $$ x=128 $$.
The true solution is $$ x=128 $$.

What is the true solution to the logarithmic equation below?

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