Solve 0.5*10^-(n-1)=0.001

$$0.5 \times 10 ^ { - ( n - 1 ) } = 0.001$$

Solve for n

$$n=\log_{10}\left(5000\right)\approx 3.698970004$$

Steps Using Definition of Logarithm

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

$$0.5\times 10^{-\left(n-1\right)}=0.001$$

Multiply both sides by 2.

$$10^{-\left(n-1\right)}=0.002$$

Take the logarithm of both sides of the equation.

$$\log(10^{-\left(n-1\right)})=\log(0.002)$$

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

$$-\left(n-1\right)\log(10)=\log(0.002)$$

Divide both sides by -1.

$$n-1=-\frac{\log(500)}{-1}$$

Add 1 to both sides of the equation.

$$n=\log(500)-\left(-1\right)$$