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:

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

The solution set is {-0.922}