Solve 6=1.2^n-1

$$6 = 1.2 ^ { n - 1 }$$

Solve for n

$$n=-\log_{1.2}\left(\frac{5}{36}\right)\approx 10.82746912$$

Steps Using Definition of Logarithm

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

$$1.2^{n-1}=6$$

Take the logarithm of both sides of the equation.

$$\log(1.2^{n-1})=\log(6)$$

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

$$\left(n-1\right)\log(1.2)=\log(6)$$

Divide both sides by \log(1.2).

$$n-1=\frac{\log(6)}{\log(1.2)}$$

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

$$n-1=\log_{1.2}\left(6\right)$$

Add 1 to both sides of the equation.

$$n=\frac{\ln(6)}{\ln(\frac{6}{5})}-\left(-1\right)$$