Solve 0.3=6.3*(0.9)^n

$$0.3 = 6.3 \times ( 0.9 ) ^ { n }$$

Solve for n

$$n=-\log_{0.9}\left(21\right)\approx 28.896237065$$

Steps Using Definition of Logarithm

Divide both sides by 6.3.

$$\frac{0.3}{6.3}=0.9^{n}$$

Expand \frac{0.3}{6.3} by multiplying both numerator and the denominator by 10.

$$\frac{3}{63}=0.9^{n}$$

Reduce the fraction \frac{3}{63} to lowest terms by extracting and canceling out 3.

$$\frac{1}{21}=0.9^{n}$$

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

$$0.9^{n}=\frac{1}{21}$$

Take the logarithm of both sides of the equation.

$$\log(0.9^{n})=\log(\frac{1}{21})$$

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

$$n\log(0.9)=\log(\frac{1}{21})$$

Divide both sides by \log(0.9).

$$n=\frac{\log(\frac{1}{21})}{\log(0.9)}$$

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

$$n=\log_{0.9}\left(\frac{1}{21}\right)$$

Solve for n (complex solution)

$$n=\frac{i\times 2\pi n_{1}}{\ln(0.9)}-\log_{0.9}\left(21\right)$$

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