TMUA 2016 Paper 2 Q7 — Worked Solution

P2·Indices, Logarithms & Exponentials·6.5Official · 2016 · Q7

proof-constructionlogarithm-lawsstep-ordering

The four real numbers

a

b

c

, and

d

are all greater than 1. Suppose that they satisfy the equation

lo g_{c} d = (lo g_{a} b)^{2}

. Use some of the lines given to construct a proof that, in this case, it follows that

(*) lo g_{b} d = (lo g_{a} b) (lo g_{a} c) .

The lines:
(1) Let

x = lo g_{a} b

and

y = lo g_{a} c

(2)

d = (c^{x})^{2}

(3)

d = c^{(x^{2})}

(4)

d = b^{x y}

(5)

d = (a^{y})^{(x^{2})}

(6)

d = ((a^{y})^{x})^{2}

(7)

d = (a^{x})^{x y}

(8)

d = a^{(y^{2} x)}

(9)

d = a^{(x^{2} y)}

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