Equation \(\displaystyle{\log{{\left({x}^{{2}}+{2}{a}{x}\right)}}}={\log{{\left({4}{x}-{4}{a}-{13}\right)}}}\) has only one solution; then

Anika Boyd

Anika Boyd

Answered question

2022-04-05

Equation log(x2+2ax)=log(4x4a13) has only one solution; then exhaustive set of values of a is

Answer & Explanation

Laylah Hebert

Laylah Hebert

Beginner2022-04-06Added 15 answers

Hint: aloga(x)=x, so:
x2+2ax=4x4a13
x2+2(a2)x+(4a+13)=0
kaosimqu5t

kaosimqu5t

Beginner2022-04-07Added 10 answers

log(x2+2ax)=log(4x4a13)
This implies that
x2+2ax=4x4a13
or
x2+2ax4x+4a+13=0
or
x2+(2a4)x+(4a+13)=0
Since the equation has just one solution instead of the usual two distinct solutions, then the two solutions must be same i.e. discriminant =0.
Hence we get that
(2a4)2=41(4a+13)
or
4a216a+16=16a+52
or
4a232a36=0
or
a28a9=0
or
(a9)(a+1)=0
So the values of a are 1 and 9

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