Solve 0.133333=0.75^n-1

$$0.133333 = 0.75 ^ { n - 1 }$$

Solve for n

$$n=-\log_{0.75}\left(\frac{4000000}{399999}\right)\approx 8.00393147$$

Steps Using Definition of Logarithm

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

$$0.75^{n-1}=0.133333$$

Take the logarithm of both sides of the equation.

$$\log(0.75^{n-1})=\log(0.133333)$$

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

$$\left(n-1\right)\log(0.75)=\log(0.133333)$$

Divide both sides by \log(0.75).

$$n-1=\frac{\log(0.133333)}{\log(0.75)}$$

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

$$n-1=\log_{0.75}\left(0.133333\right)$$

Add 1 to both sides of the equation.

$$n=\frac{\ln(\frac{133333}{1000000})}{\ln(\frac{3}{4})}-\left(-1\right)$$