Let a, b, c>0 and a ≠ 1. The equation _b((b^x)^x) + _a≤ft(\;c^x/b^x) + _a≤ft(\;1/b) _a(c)=0 has a repeated root when

Correct Answer is : c= 1bc=\;1/bc=b1

Correct Answer is : c= 1bc=\;1/bc=b1

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Oxford University MAT 2017 Solved Papers

Question : 9 of 10

Marks: +1, -0

Let

a, b, c>0

a \neq 1

\log_b((b^x)^x) + \log_a\left(\;\frac{c^x}{b^x}\right) + \log_a\left(\;\frac{1}{b}\right) \log_a(c)=0

has a repeated root when

Solution:

By algebraic manipulation we have

\log_b((b^x)^x) + \log_a\left(\;\frac{c^x}{b^x}\right) + \log_a\left(\;\frac{1}{b}\right) \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\left(\;\frac{1}{c}\right) \;\; =\log_a(b)

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