Solve 2^1-x=3

$$2 ^ { 1 - x } = 3$$

Solve for x

$$x=-\log_{2}\left(3\right)+1\approx -0.584962501$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$2^{-x+1}=3$$

Take the logarithm of both sides of the equation.

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

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\left(-x+1\right)\log(2)=\log(3)$$

Divide both sides by \log(2).

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

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$-x+1=\log_{2}\left(3\right)$$

Subtract 1 from both sides of the equation.

$$-x=\log_{2}\left(3\right)-1$$

Divide both sides by -1.

$$x=\frac{\log_{2}\left(3\right)-1}{-1}$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(2)}+\log_{2}\left(\frac{2}{3}\right)$$

$$n_{1}\in \mathrm{Z}$$