Solve e^2x+4=1

$$e ^ { 2 x + 4 } = 1$$

Solve for x

$$x=-2$$

Steps Using Definition of Logarithm

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

$$e^{2x+4}=1$$

Take the logarithm of both sides of the equation.

$$\log(e^{2x+4})=\log(1)$$

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

$$\left(2x+4\right)\log(e)=\log(1)$$

Divide both sides by \log(e).

$$2x+4=\frac{\log(1)}{\log(e)}$$

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

$$2x+4=\log_{e}\left(1\right)$$

Subtract 4 from both sides of the equation.

$$2x=-4$$

Divide both sides by 2.

$$x=-\frac{4}{2}$$

Solve for x (complex solution)

$$x=\pi n_{1}i-2$$

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