Solve e^-4x+1=5

$$e ^ { - 4 x + 1 } = 5$$

Solve for x

$$x=\frac{1-\ln(5)}{4}\approx -0.152359478$$

Steps Using Definition of Logarithm

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

$$e^{-4x+1}=5$$

Take the logarithm of both sides of the equation.

$$\log(e^{-4x+1})=\log(5)$$

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

$$\left(-4x+1\right)\log(e)=\log(5)$$

Divide both sides by \log(e).

$$-4x+1=\frac{\log(5)}{\log(e)}$$

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

$$-4x+1=\log_{e}\left(5\right)$$

Subtract 1 from both sides of the equation.

$$-4x=\ln(5)-1$$

Divide both sides by -4.

$$x=\frac{\ln(5)-1}{-4}$$

Solve for x (complex solution)

$$x=-\frac{i\pi n_{1}}{2}-\frac{\ln(5)}{4}+\frac{1}{4}$$

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