Solve 60e^-0.001x=30

$$60 e ^ { - 0.001 x } = 30$$

Solve for x

$$x=1000\ln(2)\approx 693.14718056$$

Steps Using Definition of Logarithm

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

$$60e^{-0.001x}=30$$

Divide both sides by 60.

$$e^{-0.001x}=\frac{1}{2}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.001x})=\log(\frac{1}{2})$$

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

$$-0.001x\log(e)=\log(\frac{1}{2})$$

Divide both sides by \log(e).

$$-0.001x=\frac{\log(\frac{1}{2})}{\log(e)}$$

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

$$-0.001x=\log_{e}\left(\frac{1}{2}\right)$$

Multiply both sides by -1000.

$$x=-\frac{\ln(2)}{-0.001}$$

Solve for x (complex solution)

$$x=-i\times 2000\pi n_{1}+1000\ln(2)$$

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