Solve 0.3=(0.8)^n

$$0.3 = ( 0.8 ) ^ { n }$$

Solve for n

$$n=\log_{0.8}\left(0.3\right)\approx 5.395507946$$

Steps Using Definition of Logarithm

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

$$0.8^{n}=0.3$$

Take the logarithm of both sides of the equation.

$$\log(0.8^{n})=\log(0.3)$$

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

$$n\log(0.8)=\log(0.3)$$

Divide both sides by \log(0.8).

$$n=\frac{\log(0.3)}{\log(0.8)}$$

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

$$n=\log_{0.8}\left(0.3\right)$$

Solve for n (complex solution)

$$n=\frac{i\times 2\pi n_{1}}{\ln(0.8)}+\log_{0.8}\left(0.3\right)$$

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