Solve 0.42*100^k=42

$$0.42 \times 100 ^ { k } = 42$$

Solve for k

$$k=1$$

Steps Using Definition of Logarithm

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

$$0.42\times 100^{k}=42$$

Divide both sides of the equation by 0.42, which is the same as multiplying both sides by the reciprocal of the fraction.

$$100^{k}=100$$

Take the logarithm of both sides of the equation.

$$\log(100^{k})=\log(100)$$

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

$$k\log(100)=\log(100)$$

Divide both sides by \log(100).

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

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

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