Solve 0.00032=0.2^x

$$0.00032= { 0.2 }^{ x }$$

Solve for x

$$x=5$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$0.2^{x}=0.00032$$

Take the logarithm of both sides of the equation.

$$\log(0.2^{x})=\log(0.00032)$$

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

$$x\log(0.2)=\log(0.00032)$$

Divide both sides by \log(0.2).

$$x=\frac{\log(0.00032)}{\log(0.2)}$$

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

$$x=\log_{0.2}\left(0.00032\right)$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{\ln(5)}+5$$

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