Solve e^3-4x=4

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

Solve for x

$$x=-\frac{\ln(2)}{2}+\frac{3}{4}\approx 0.40342641$$

Steps Using Definition of Logarithm

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

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

Take the logarithm of both sides of the equation.

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

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

$$\left(-4x+3\right)\log(e)=\log(4)$$

Divide both sides by \log(e).

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

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

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

Subtract 3 from both sides of the equation.

$$-4x=2\ln(2)-3$$

Divide both sides by -4.

$$x=\frac{2\ln(2)-3}{-4}$$

Solve for x (complex solution)

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

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