Question

Select the correct choice below and fill in the blank if necessary

Answer 1

The given equation is:

`2^(x-2)=3^(2x)`

Take the logarithm of both sides and apply the exponent rule:

`\begin{gathered} \log_{\placeholder{⬚}}2^(x-2)=\log_{\placeholder{⬚}}3^(2x) \\ (x-2)\log_{\placeholder{⬚}}2=(2x)\log_{\placeholder{⬚}}3 \end{gathered}`

Apply the distributive property:

`x\log_{\placeholder{⬚}}2-2\log_{\placeholder{⬚}}2=2x\log_{\placeholder{⬚}}3`

Reorder the equation and subtract xlog2 from both sides:

`\begin{gathered} 2xlog3-xlog2=xlog2-2log2-xlog2 \\ 2x\log_{\placeholder{⬚}}3-x\log_{\placeholder{⬚}}2=-2\log_{\placeholder{⬚}}2 \end{gathered}`

Factorize x:

`x(2\log_{\placeholder{⬚}}3-\log_{\placeholder{⬚}}2)=-2\log_{\placeholder{⬚}}2`

Solve for x:

`\begin{gathered} x=\frac{-2\log_{\placeholder{⬚}}2}{2\log_{\placeholder{⬚}}3-\log_{\placeholder{⬚}}2} \\ x=(-2*0.301)/(2*0.477-0.301) \\ x=(-0.602)/(0.954-0.301) \\ x=(-0.602)/(0.653) \\ x=-0.922 \end{gathered}`

The solution set is {-0.922}