Solve 0.4^x=0.01

$${ 0.4 }^{ x } =0.01$$

Solve for x

$$x=-2\log_{0.4}\left(10\right)\approx 5.025883189$$

Steps Using Definition of Logarithm

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

$$0.4^{x}=0.01$$

Take the logarithm of both sides of the equation.

$$\log(0.4^{x})=\log(0.01)$$

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

$$x\log(0.4)=\log(0.01)$$

Divide both sides by \log(0.4).

$$x=\frac{\log(0.01)}{\log(0.4)}$$

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

$$x=\log_{0.4}\left(0.01\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.4)}-2\log_{0.4}\left(10\right)$$

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