2.19: Simultaneous logarithmic equations leading to a cubic

0606/23/O/N/20 — Question 8 · 10 marks · Equations, Log laws

Do not use a calculator in this question.

log⁡2(y+1)=3−2log⁡2x\log_{2}(y + 1) = 3 - 2\log_{2} xlog⁡2(x+2)=2+log⁡2y\log_{2}(x + 2) = 2 + \log_{2} y
(a)
Show that x3+6x2−32=0x^{3} + 6x^{2} - 32 = 0.
[4]
(b)
Find the roots of x3+6x2−32=0x^{3} + 6x^{2} - 32 = 0.
[4]
(c)
Give a reason why only one root is a valid solution of the logarithmic equations. Find the value of yy corresponding to this root.
[2]