Solve 1000001=50000(1+0.08)^n

$$1000001 = 50000 ( 1 + 0.08 ) ^ { n }$$

Solve for n

$$n=\log_{1.08}\left(20.00002\right)\approx 38.925321557$$

Steps Using Definition of Logarithm

Divide both sides by 50000.

$$\frac{1000001}{50000}=\left(1+0.08\right)^{n}$$

Add 1 and 0.08 to get 1.08.

$$\frac{1000001}{50000}=1.08^{n}$$

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

$$1.08^{n}=\frac{1000001}{50000}$$

Take the logarithm of both sides of the equation.

$$\log(1.08^{n})=\log(\frac{1000001}{50000})$$

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

$$n\log(1.08)=\log(\frac{1000001}{50000})$$

Divide both sides by \log(1.08).

$$n=\frac{\log(\frac{1000001}{50000})}{\log(1.08)}$$

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

$$n=\log_{1.08}\left(\frac{1000001}{50000}\right)$$