Evaluate the following: i) log5(3/2) + 3 log5(15/2) - log5(81/8) ii) 1/3 log2 64

$$\log_5\left(\frac{3}{2}\right) + \log_5\left(\left(\frac{15}{2}\right)^3\right) - \log_5\left(\frac{81}{8}\right)$$ $$\log_5\left(\frac{3}{2}\right) + \log_5\left(\frac{15^3}{2^3}\right) - \log_5\left(\frac{81}{8}\right)$$ $$\log_5\left(\frac{3}{2}\right) + \log_5\left(\frac{3375}{8}\right) - \log_5\left(\frac{81}{8}\right)$$

$$\log_5\left(\frac{3 \times 3375}{2 \times 8} \times \frac{8}{81}\right)$$ $$\log_5\left(\frac{3 \times 3375}{2 \times 81}\right)$$ $$\log_5\left(\frac{10125}{162}\right)$$

Divide both numerator and denominator by common factors. $10125 \div 81 = 125$.

$$\log_5\left(\frac{125}{2}\right)$$

This cannot be simplified further to a simple integer or fraction without using a calculator. However, if the intention was to get a simpler number, let's recheck the calculation.
Let's re-evaluate the simplification:

$$\log_5\left(\frac{3}{2}\right) + \log_5\left(\frac{15^3}{2^3}\right) - \log_5\left(\frac{81}{8}\right)$$ $$\log_5\left(\frac{3}{2} \times \frac{15^3}{2^3} \times \frac{8}{81}\right)$$ $$\log_5\left(\frac{3}{2} \times \frac{3375}{8} \times \frac{8}{81}\right)$$ $$\log_5\left(\frac{3 \times 3375 \times 8}{2 \times 8 \times 81}\right)$$

Cancel out the 8s:

$$\log_5\left(\frac{3 \times 3375}{2 \times 81}\right)$$ $$\log_5\left(\frac{10125}{162}\right)$$

Divide by 3: $10125 \div 3 = 3375$, $162 \div 3 = 54$.

$$\log_5\left(\frac{3375}{54}\right)$$

Divide by 3 again: $3375 \div 3 = 1125$, $54 \div 3 = 18$.

$$\log_5\left(\frac{1125}{18}\right)$$

Divide by 3 again: $1125 \div 3 = 375$, $18 \div 3 = 6$.

$$\log_5\left(\frac{375}{6}\right)$$

Divide by 3 again: $375 \div 3 = 125$, $6 \div 3 = 2$.

$$\log_5\left(\frac{125}{2}\right)$$

This is the most simplified form.

The final answer for i) is $\boxed{\log_5\left(\frac{125}{2}\right)}$.

This is the cube root of 64.

$$64^{\frac{1}{3}} = \sqrt[3]{64} = 4$$

$$2^x = 4 \implies x = 2$$ $$\log_2(4) = 2$$

The final answer for ii) is $\boxed{2}$.