How do you solve $log _ 5 (x-3)+log _ 5(x+1) = log _ 5(x+3)$ ?

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Answer 1 · 2015-12-13T22:19:19

$x=(3+sqrt33)/2$

Combine using logarithm rules.

$log_5((x-3)(x+1))=log_5(x+3)$

Raise both sides to the $5$ th power.

$5^(log_5((x-3)(x+1)))=5^(log_5(x+3))$

$(x-3)(x+1)=x+3$

$x^2-2x-3=x+3$

$x^2-3x-6=0$

$x=(3+-sqrt(9+24))/2=(3+-sqrt33)/2$

Throw out the negative answer. It cannot be used because a logarithm has to be a function of a POSITIVE number.

Thus, $x=(3+sqrt33)/2$ .

How do you solve log _ 5 (x-3)+log _ 5(x+1) = log _ 5(x+3)log _ 5 (x-3)+log _ 5(x+1) = log _ 5(x+3)?