Solve 2=1*1.05^x

$$2=1 \times { 1.05 }^{ x }$$

Solve for x

$$x=\log_{1.05}\left(2\right)\approx 14.206699083$$

Steps Using Definition of Logarithm

Divide both sides by 1.

$$2=1.05^{x}$$

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

$$1.05^{x}=2$$

Take the logarithm of both sides of the equation.

$$\log(1.05^{x})=\log(2)$$

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

$$x\log(1.05)=\log(2)$$

Divide both sides by \log(1.05).

$$x=\frac{\log(2)}{\log(1.05)}$$

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

$$x=\log_{1.05}\left(2\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.05)}+\log_{1.05}\left(2\right)$$

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