Solve 0.5=0.7937^t

$$0.5 = 0.7937 ^ { t }$$

Solve for t

$$t=-\log_{0.7937}\left(2\right)\approx 2.999991395$$

Steps Using Definition of Logarithm

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

$$0.7937^{t}=0.5$$

Take the logarithm of both sides of the equation.

$$\log(0.7937^{t})=\log(0.5)$$

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

$$t\log(0.7937)=\log(0.5)$$

Divide both sides by \log(0.7937).

$$t=\frac{\log(0.5)}{\log(0.7937)}$$

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

$$t=\log_{0.7937}\left(0.5\right)$$

Solve for t (complex solution)

$$t=\frac{i\times 2\pi n_{1}}{\ln(0.7937)}-\log_{0.7937}\left(2\right)$$

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