Solve 5000=30000*2^t

$$5000 = 30000 \cdot 2 ^ { t }$$

Solve for t

$$t=-\log_{2}\left(6\right)\approx -2.584962501$$

Steps Using Definition of Logarithm

Divide both sides by 30000.

$$\frac{5000}{30000}=2^{t}$$

Reduce the fraction \frac{5000}{30000} to lowest terms by extracting and canceling out 5000.

$$\frac{1}{6}=2^{t}$$

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

$$2^{t}=\frac{1}{6}$$

Take the logarithm of both sides of the equation.

$$\log(2^{t})=\log(\frac{1}{6})$$

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

$$t\log(2)=\log(\frac{1}{6})$$

Divide both sides by \log(2).

$$t=\frac{\log(\frac{1}{6})}{\log(2)}$$

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

$$t=\log_{2}\left(\frac{1}{6}\right)$$

Solve for t (complex solution)

$$t=\frac{2\pi n_{1}i}{\ln(2)}-\log_{2}\left(6\right)$$

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