TMUA Official 2016 Paper 2 · Question 7

The four real numbers $a,b,c$ , and $d$ are all greater than 1.

Suppose that they satisfy the equation $\log_c d=(\log_a b)^2$ .

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

(*)\quad\log_b d=(\log_a b)(\log_a c).

A(1). Then (2), so (6), so (8), so (7), and therefore (4), hence (*) as required.

B(1). Then (2), so (7), so (8), so (6), and therefore (4), hence (*) as required.

C(1). Then (3), so (5), so (9), so (7), and therefore (4), hence (*) as required.

D(1). Then (3), so (7), so (9), so (5), and therefore (4), hence (*) as required.

E(1). Then (4), so (5), so (9), so (7), and therefore (3), hence (*) as required.

F(1). Then (4), so (6), so (8), so (7), and therefore (2), hence (*) as required.

G(1). Then (4), so (7), so (8), so (6), and therefore (2), hence (*) as required.

H(1). Then (4), so (7), so (9), so (5), and therefore (3), hence (*) as required.

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