Solve 1.05^t=2

$${ 1.05 }^{ t } = 2$$

Solve for t

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

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$1.05^{t}=2$$

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(1.05).

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

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

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

Solve for t (complex solution)

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

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