Oxford University MAT 2017 Solved Papers

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Question : 9 of 10

Marks: +1, -0

Let

a, b, c>0

and

a ≠ 1

. The equation

\log _b((b^x)^x)+\log _a(\;{c^x}/{b^x})+\log _a(\;{1}/{b}) \log _a(c)=0

has a repeated root when

$b^2=4 a c$
$b=\;{1}/{a}$
$c=\;{b}/{a}$
$c=\;{1}/{b}$
$a=b=c$

Solution:

By algebraic manipulation we have

\log _b((b^x)^x)+\log _a(\;{c^x}/{b^x})+\log _a(\;{1}/{b}) \log _a(c) \;\; =0

x^2+x(\log _a(c)-\log _a(b))-\log _a(b) \log _a(c) \;\; =0

(x+\log _a(c))(x-\log _a(b)) \;\; =0 .

The question asks when the equation has a repeated root, which is satisfied when

-\log _a(c) \;\; =\log _a(b)

\log _a(\;{1}/{c}) \;\; =\log _a(b)

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