Question

Solve.2^x-3 = (1/2)^2x-3Answer additional questions in the Show Your Work space.What is the first step to solving the equation?

Answer 1

Answer:

x = 2

First step: Apply logarithm in both sides

Explanation:

The given expression is

$2^{x-3}=(\frac{1}{2})^{2x-3}$

Then, the first step is to apply logarithm in base 2 to both sides, so

$\begin{gathered} \log _22^{x-3}=\log _2(\frac{1}{2})^{2x-3}_{^{}} \\ (x-3)\log _22=(2x-3)\log _2(\frac{1}{2}) \end{gathered}$

Then, we can calculate the logarithms and solve for x, so

$\begin{gathered} (x-3)(1)=(2x-3)(-1) \\ x-3=2x(-1)-3(-1) \\ x-3=-2x+3 \end{gathered}$$\begin{gathered} x-3+3=-2x+3+3 \\ x=-2x+6 \\ x+2x=-2x+6+2x \\ 3x=6 \\ \frac{3x}{3}=\frac{6}{3} \\ x=2 \end{gathered}$

Therefore, the solution is x = 2