Solve 3=2+5e^-4x

$$3 = 2 + 5 e ^ { - 4 x }$$

Solve for x

$$x=\frac{\ln(5)}{4}\approx 0.402359478$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$2+5e^{-4x}=3$$

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

$$5e^{-4x}+2=3$$

Subtract 2 from both sides of the equation.

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

Divide both sides by 5.

$$e^{-4x}=\frac{1}{5}$$

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(e).

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

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

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

Divide both sides by -4.

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

Solve for x (complex solution)

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

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