Solve 600=0.2^n-1

$$600 = 0.2 ^ { n - 1 }$$

Solve for n

$$n=-\log_{5}\left(24\right)-1\approx -2.974635869$$

Steps Using Definition of Logarithm

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

$$0.2^{n-1}=600$$

Take the logarithm of both sides of the equation.

$$\log(0.2^{n-1})=\log(600)$$

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

$$\left(n-1\right)\log(0.2)=\log(600)$$

Divide both sides by \log(0.2).

$$n-1=\frac{\log(600)}{\log(0.2)}$$

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

$$n-1=\log_{0.2}\left(600\right)$$

Add 1 to both sides of the equation.

$$n=-\log_{5}\left(600\right)-\left(-1\right)$$

Solve for n (complex solution)

$$n=-\frac{i\times 2\pi n_{1}}{\ln(5)}-\log_{5}\left(120\right)$$

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