Let $$a = 3\sqrt{2}$$ and $$b = \frac{1}{5^{1/6}\sqrt{6}}$$. If $$x, y \in \mathbb{R}$$ are such that $$3x + 2y = \log_a (18)^{5/4}$$ and $$2x - y = \log_b (\sqrt{1080})$$, then $$4x + 5y$$ is equal to ________.

The two unknowns $$x$$ and $$y$$ satisfy the linear system

$$3x + 2y = \log_a \left(18\right)^{5/4} \qquad \text{and} \qquad 2x - y = \log_b \left(\sqrt{1080}\right)$$

First simplify each logarithm.

1. Computing $$\log_a \left(18\right)^{5/4}$$

The base is $$a = 3\sqrt{2}=3\cdot 2^{1/2}=3^{1}\, 2^{1/2}$$.
The argument is $$\left(18\right)^{5/4}=(3^{2}\,2^{1})^{5/4}=3^{5/2}\,2^{5/4}$$.

Using $$\log_c d = \dfrac{\ln d}{\ln c}$$,

$$\log_a\left(18\right)^{5/4}= \dfrac{(5/2)\ln 3 + (5/4)\ln 2}{\ln 3 + (1/2)\ln 2}$$

Factor out $$\dfrac54$$ in the numerator and $$\dfrac12$$ in the denominator:

$$= \dfrac{(5/4)(2\ln 3 +\ln 2)}{(1/2)(2\ln 3 +\ln 2)} = \dfrac{5/4}{1/2}= \dfrac52$$

Hence

$$3x+2y = \dfrac52 \qquad -(1)$$

2. Computing $$\log_b \left(\sqrt{1080}\right)$$

The base is $$b=\dfrac1{5^{1/6}\sqrt6}=5^{-1/6}\,2^{-1/2}\,3^{-1/2}$$,
so $$\ln b = -\!\left(\tfrac16\ln5+\tfrac12\ln2+\tfrac12\ln3\right)=-M \text{ where } M=\tfrac16\ln5+\tfrac12\ln2+\tfrac12\ln3$$.

Factorise the argument: $$1080 = 2^{3}\,3^{3}\,5$$, therefore

$$\sqrt{1080}=1080^{1/2}=2^{3/2}\,3^{3/2}\,5^{1/2}$$

and

$$\ln\!\left(\sqrt{1080}\right)=\tfrac32\ln2+\tfrac32\ln3+\tfrac12\ln5 =(1/2)(3\ln2+3\ln3+\ln5)=N_1.$$

Compute the ratio:

$$\dfrac{N_1}{M} = \dfrac{(1/2)(3\ln2+3\ln3+\ln5)} {\,\tfrac16\ln5+\tfrac12\ln2+\tfrac12\ln3\,} =\dfrac{9\ln2+9\ln3+3\ln5}{\ln5+3\ln2+3\ln3}=3$$

Therefore $$\ln\!\left(\sqrt{1080}\right)=3M$$ and

$$\log_b\!\left(\sqrt{1080}\right)=\dfrac{\ln(\sqrt{1080})}{\ln b} =\dfrac{3M}{-M}=-3$$

Thus

$$2x - y = -3 \qquad -(2)$$

3. Solving the linear system

From (2): $$y = 2x + 3$$.

Substitute into (1):

$$3x + 2(2x+3)=\dfrac52$$

$$3x + 4x + 6 = \dfrac52$$

$$7x = \dfrac52 - 6 = \dfrac52 - \dfrac{12}2 = -\dfrac72$$

$$x = -\dfrac72 \div 7 = -\dfrac12$$

Then $$y = 2(-\tfrac12) + 3 = -1 + 3 = 2$$.

4. Required expression

$$4x + 5y = 4\!\left(-\dfrac12\right) + 5(2) = -2 + 10 = 8$$

Hence, $$4x + 5y = 8$$.

Answer: 8