Solve -14+3e^-x=11

$$- 14 + 3 e ^ { - x } = 11$$

Solve for x

$$x=\ln(\frac{3}{25})\approx -2.120263536$$

Steps Using Definition of Logarithm

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

$$3e^{-x}-14=11$$

Add 14 to both sides of the equation.

$$3e^{-x}=25$$

Divide both sides by 3.

$$e^{-x}=\frac{25}{3}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-x})=\log(\frac{25}{3})$$

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

$$-x\log(e)=\log(\frac{25}{3})$$

Divide both sides by \log(e).

$$-x=\frac{\log(\frac{25}{3})}{\log(e)}$$

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

$$-x=\log_{e}\left(\frac{25}{3}\right)$$

Divide both sides by -1.

$$x=\frac{\ln(\frac{25}{3})}{-1}$$

Solve for x (complex solution)

$$x=-2\pi n_{1}i+\ln(\frac{3}{25})$$

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