Solve 3e^4x+2=11

$$3 e ^ { 4 x } + 2 = 11$$

Solve for x

$$x=\frac{\ln(3)}{4}\approx 0.274653072$$

Steps Using Definition of Logarithm

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

$$3e^{4x}+2=11$$

Subtract 2 from both sides of the equation.

$$3e^{4x}=9$$

Divide both sides by 3.

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

Take the logarithm of both sides of the equation.

$$\log(e^{4x})=\log(3)$$

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

$$4x\log(e)=\log(3)$$

Divide both sides by \log(e).

$$4x=\frac{\log(3)}{\log(e)}$$

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

$$4x=\log_{e}\left(3\right)$$

Divide both sides by 4.

$$x=\frac{\ln(3)}{4}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2}+\frac{\ln(3)}{4}$$

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