Solve e^-9x=7.2

$$e ^ { - 9 x } = 7.2$$

Solve for x

$$x=\frac{\ln(\frac{5}{36})}{9}\approx -0.219342336$$

Steps Using Definition of Logarithm

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

$$e^{-9x}=7.2$$

Take the logarithm of both sides of the equation.

$$\log(e^{-9x})=\log(7.2)$$

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

$$-9x\log(e)=\log(7.2)$$

Divide both sides by \log(e).

$$-9x=\frac{\log(7.2)}{\log(e)}$$

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

$$-9x=\log_{e}\left(7.2\right)$$

Divide both sides by -9.

$$x=\frac{\ln(\frac{36}{5})}{-9}$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{9}+\frac{\ln(\frac{5}{36})}{9}$$

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