Solve 80000000=20001.2^x

$$80000000 = 2000 { 1.2 }^{ x }$$

Solve for x

$$x=2\log_{1.2}\left(200\right)\approx 58.12058058$$

Steps Using Definition of Logarithm

Divide both sides by 2000.

$$\frac{80000000}{2000}=1.2^{x}$$

Divide 80000000 by 2000 to get 40000.

$$40000=1.2^{x}$$

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

$$1.2^{x}=40000$$

Take the logarithm of both sides of the equation.

$$\log(1.2^{x})=\log(40000)$$

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

$$x\log(1.2)=\log(40000)$$

Divide both sides by \log(1.2).

$$x=\frac{\log(40000)}{\log(1.2)}$$

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

$$x=\log_{1.2}\left(40000\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.2)}+2\log_{1.2}\left(200\right)$$

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