Solve 500=2^t/3

$$500 = 2 ^ { t / 3 }$$

Solve for t

$$t=9\log_{2}\left(5\right)+6\approx 26.897352854$$

Steps Using Definition of Logarithm

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

$$2^{\frac{t}{3}}=500$$

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

$$2^{\frac{1}{3}t}=500$$

Take the logarithm of both sides of the equation.

$$\log(2^{\frac{1}{3}t})=\log(500)$$

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

$$\frac{1}{3}t\log(2)=\log(500)$$

Divide both sides by \log(2).

$$\frac{1}{3}t=\frac{\log(500)}{\log(2)}$$

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

$$\frac{1}{3}t=\log_{2}\left(500\right)$$

Multiply both sides by 3.

$$t=\frac{\log_{2}\left(500\right)}{\frac{1}{3}}$$