7. (i) Use logarithms to solve the equation $8^{2x+1} = 24$, giving your answer to 3 decimal places - Edexcel - A-Level Maths Pure - Question 8 - 2015 - Paper 2

To find $y$, we start from the equation:

$ext{log}_{2}(11y - 3) - ext{log}_{2}(3) - 2 ext{log}_{2}(y) = 1$

Using properties of logarithms, we can simplify it to:

$ext{log}_{2}(\frac{11y - 3}{3y^{2}}) = 1$

This means:

$\frac{11y - 3}{3y^{2}} = 2$

Multiplying both sides by $3y^{2}$ gives us:

$11y - 3 = 6y^{2}$

Rearranging leads to:

$6y^{2} - 11y + 3 = 0$

We can now apply the quadratic formula,

$y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} = \frac{11 \pm \sqrt{121 - 72}}{12} = \frac{11 \pm \sqrt{49}}{12}$

Calculating the roots:

$y = \frac{11 + 7}{12} = 1.5, \quad y = \frac{11 - 7}{12} = \frac{1}{3}$

Considering the constraint $y > \frac{3}{11}$, we find the valid solution:

$y = 1.5$