Solve 2(3+e^1+4x)=8

$$2 ( 3 + e ^ { 1 + 4 x } ) = 8$$

Solve for x

$$x=-\frac{1}{4}=-0.25$$

Steps Using Definition of Logarithm

Divide both sides by 2.

$$3+e^{1+4x}=\frac{8}{2}$$

Divide 8 by 2 to get 4.

$$3+e^{1+4x}=4$$

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

$$e^{4x+1}+3=4$$

Subtract 3 from both sides of the equation.

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

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(e).

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

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

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

Subtract 1 from both sides of the equation.

$$4x=-1$$

Divide both sides by 4.

$$x=-\frac{1}{4}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2}-\frac{1}{4}$$

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