solve log(5x-1)+log(x+5)=1

Question

Please help me solve this problem.

Answer 1

Given :

log (5x - 1) + log (x + 5) = 1

Apply property : log (a ) + log (b ) = log (ab ).

log (5x - 1)(x + 5) = 1

Distribute the terms using distributive property.

log (5x ^2 - x + 25x - 5) = 1

Substitute 1 = log (10).

log (5x ^2 - x + 25x - 5) = log (10)

Apply poperty : log (a ) = log(b ) ⇔ a = b.

5x ^2 - x + 25x - 5 = 10

Subtract 10 from each side.

5x ^2 - x + 25x - 5 - 10 = 0

5x ^2 + 24x - 15 = 0.

5x ^2 + 24x - 15 is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

x = [ - b ± sqrt (b ^2 - 4ac )] / 2a.

Substitute b = 24, a = 5, and c = - 15.

x = [ - 24 ± sqrt((24)^2 - 4 * 5 * ()- 15)] / 2 * 5

x = [- 24 ± sqrt(576 + 300)] / 10

x = [- 24 ± sqrt(876)] / 10

x = [- 24 ± 29.5] / 10

x = [- 24 + 29.5] / 10 (or) x = [- 24 - 29.5] / 10

x = 0.55 (or) x = - 5.35.

But by the definition of ln (log), 5x - 1 , x + 5 must be positive.
Hence, x > 1/5, x > -5. Therefore, x must be greater than 1/5 and
the solution -5.35 is invalid. The only solution is x = 0.55.

Solution of the equation is x = 0.55.