3.18: Logarithmic equations and identities

0606/21/M/J/21 — Question 8 · 8 marks

In this question, aa, bb, cc and dd are positive constants.
(a)(i)
It is given that y=log⁡a(x+3)+log⁡a(2x−1)y=\log_{a}(x+3)+\log_{a}(2x-1). Explain why xx must be greater than 12\tfrac{1}{2}.
[1]
(a)(ii)
Find the exact solution of the equation log⁡a6log⁡a(y+3)=2\dfrac{\log_{a}6}{\log_{a}(y+3)}=2.
[3]
(b)
Write the expression log⁡a9+(log⁡ab)(log⁡b(9a))\log_{a}9+(\log_{a}b)\bigl(\log_{\sqrt{b}}(9a)\bigr) in the form c+dlog⁡a9c+d\log_{a}9, where cc and dd are integers.
[4]