If-equation-2log-x-3-log-ax-has-only-one-solution-find-the-value-of-a-

Question Number 140730 by liberty last updated on 12/May/21

If equation 2 \log (x + 3) = \log ax

has only one solution . find the

value of a .

Answered by mr W last updated on 12/May/21

a \neq 0

a x > 0

x + 3 > 0

{(x + 3)}^{2} = a x

x^{2} + (6 - a) x + 9 = 0

Δ = {(6 - a)}^{2} - 4 \times 9 = 0

{(6 - a)}^{2} = 36

6 - a = \pm 6

a = 6 \pm 6 = 0, 12

v a l i d i s a = 12

Commented by EDWIN88 last updated on 12/May/21

a = 12 or a < 0

Commented by EDWIN88 last updated on 12/May/21

put a = - 1

2 \log (x + 3) = \log (- x)

{\begin{cases} x + 3 > 0 \Rightarrow x > - 3 \\ - x > 0 \Rightarrow x < 0 \end{cases} \Rightarrow - 3 < x < 0

\log {(x + 3)}^{2} = \log (- x)

\Rightarrow x^{2} + 6 x + 9 + x = 0

\Rightarrow x^{2} + 7 x + 9 = 0

\Rightarrow x = \frac{- 7 + \sqrt{13}}{2} \approx - 1.697

\Rightarrow x = \frac{- 7 - \sqrt{13}}{2} \approx - 5.302 \leftarrow not solution

Commented by mr W last updated on 12/May/21

y e s . y o u a r e r i g h t .

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