Solve e^3-2x=8

$$e ^ { 3 - 2 x } = 8$$

Solve for x

$$x=\frac{3-3\ln(2)}{2}\approx 0.460279229$$

Steps Using Definition of Logarithm

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

$$e^{-2x+3}=8$$

Take the logarithm of both sides of the equation.

$$\log(e^{-2x+3})=\log(8)$$

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

$$\left(-2x+3\right)\log(e)=\log(8)$$

Divide both sides by \log(e).

$$-2x+3=\frac{\log(8)}{\log(e)}$$

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

$$-2x+3=\log_{e}\left(8\right)$$

Subtract 3 from both sides of the equation.

$$-2x=3\ln(2)-3$$

Divide both sides by -2.

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

Solve for x (complex solution)

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

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