Solve 48e^-x30=6

$$48 e ^ { - x 30 } = 6$$

Solve for x_30

$$x_{30}=3\ln(2)\approx 2.079441542$$

Steps Using Definition of Logarithm

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

$$48e^{-x_{30}}=6$$

Divide both sides by 48.

$$e^{-x_{30}}=\frac{1}{8}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-x_{30}})=\log(\frac{1}{8})$$

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

$$-x_{30}\log(e)=\log(\frac{1}{8})$$

Divide both sides by \log(e).

$$-x_{30}=\frac{\log(\frac{1}{8})}{\log(e)}$$

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

$$-x_{30}=\log_{e}\left(\frac{1}{8}\right)$$

Divide both sides by -1.

$$x_{30}=-\frac{3\ln(2)}{-1}$$

Solve for x_30 (complex solution)

$$x_{30}=-2\pi n_{1}i+3\ln(2)$$

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