Solve (1/1000)^k=100

$$( \frac { 1 } { 1000 } ) ^ { k } = 100$$

Solve for k

$$k=-\frac{2}{3}\approx -0.666666667$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{1000}\right)^{k}=100$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{1000}\right)^{k})=\log(100)$$

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

$$k\log(\frac{1}{1000})=\log(100)$$

Divide both sides by \log(\frac{1}{1000}).

$$k=\frac{\log(100)}{\log(\frac{1}{1000})}$$

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

$$k=\log_{\frac{1}{1000}}\left(100\right)$$