Solve 1/2=1.06^t

$$\frac { 1 } { 2 } = 1.06 ^ { t }$$

Solve for t

$$t=-\log_{1.06}\left(2\right)\approx -11.895661046$$

Steps Using Definition of Logarithm

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

$$1.06^{t}=\frac{1}{2}$$

Take the logarithm of both sides of the equation.

$$\log(1.06^{t})=\log(\frac{1}{2})$$

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

$$t\log(1.06)=\log(\frac{1}{2})$$

Divide both sides by \log(1.06).

$$t=\frac{\log(\frac{1}{2})}{\log(1.06)}$$

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

$$t=\log_{1.06}\left(\frac{1}{2}\right)$$