G. Let a, b, c > 0. The equations _a b = c, \;\; _b a = c + 3/2, \;\; _c a = b,

Correct Answer is : specify a,ba, ba,b and ccc uniquely.

specify c uniquely but have infinitely many solutions for a and b.

specify c and a uniquely but have infinitely many solutions for b.

specify a and b uniquely but have infinitely many solutions for c.

have no solutions for a, b and c.

Correct Answer is : specify a,ba, ba,b and ccc uniquely.

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Oxford University MAT 2019 Solved Papers

Question : 7 of 10

Marks: +1, -0

G. Let

a, b, c > 0

\log_a b = c, \;\; \log_b a = c + \frac{3}{2}, \;\; \log_c a = b,

Solution:

The equations are equivalent to;

b = a^c, \;\; a = b^{c + \frac{3}{2}}, \;\; a = c^b

Raise each side of the second equation to the power of

c

. Then

b = b^{c^2 + \frac{3c}{2}}

, so we must have

1 = c^2 + \frac{3c}{2}

(since

b > 1

for the logarithms to make sense). This has solutions

c = -2

c = -\frac{1}{2}

, but we know

c > 0

c = \frac{1}{2}

. The remaining equations read

a = b^2

and

a = 2^{-b}

. Sketch both curves;

There is exactly one positive solution. The answer is (a)

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