The number of values of $a$ for which the equation $l o g (x^{2} + 2 a x) = l o g (8 x - 6 a - 3)$ has only one solution is

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Solution

Given $log (x^{2} + 2 a x) = log (8 x - 6 a - 3)$
$\Rightarrow x^{2} + 2 a x = 8 x - 6 a - 3$
$\Rightarrow x^{2} + (2 a - 8) x + 3 (2 a + 1) = 0$
$D = (2 a - 8)^{2} - 12 (2 a + 1)$
For one solution to exist $D = 0$
$(a - 4)^{2} - 3 (2 a + 1) = 0$
$a^{2} - 14 a + 13 = 0$
$(a - 1) (a - 13) = 0$
$a = 1, 13$
The number of values of $a$ are $2$

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