Express \(1+\lg(x^3-8)-2\lg(x-2)\), where \(x>2\), in the form \(\lg\dfrac{f(x)}{g(x)}\), where \(f(x)\) and \(g(x)\) are functions of \(x\). Give your answer in its simplest form.[3]
Solve the equation \(4\log_7(x+3)=\log_{(x+3)}7\), giving your answers in the form \(a+b\sqrt7\), where \(a\) and \(b\) are constants.[4]
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Answer:(a)(i) \((x-2)(x^2+2x+4)\). (ii) \(\lg\dfrac{10(x^2+2x+4)}{x-2}\). (b) \(-3+\sqrt7,-3+\dfrac17\sqrt7\).