Solve 2=1.065^n

$$2 = 1.065 ^ { n }$$

Solve for n

$$n=\log_{1.065}\left(2\right)\approx 11.006739041$$

Steps Using Definition of Logarithm

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

$$1.065^{n}=2$$

Take the logarithm of both sides of the equation.

$$\log(1.065^{n})=\log(2)$$

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

$$n\log(1.065)=\log(2)$$

Divide both sides by \log(1.065).

$$n=\frac{\log(2)}{\log(1.065)}$$

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

$$n=\log_{1.065}\left(2\right)$$

Solve for n (complex solution)

$$n=\frac{i\times 2\pi n_{1}}{\ln(1.065)}+\log_{1.065}\left(2\right)$$

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