Drop the logs solve for x

Question

logs x + logs (x + 6) = 10g8 (5x + 12)

Answer 1

First, we can use the properties of logarithms to simplify the equation:

log(x) + log(x + 6) = log((5x + 12)^10^8)

Using the product rule of logarithms:

log(x(x + 6)) = log((5x + 12)^10^8)

Now we can drop the logarithms:

x(x + 6) = (5x + 12)^10^8

Expand and simplify the equation:

x^2 + 6x = (5x + 12)^10^8
x^2 + 6x = 5^10^8x + 12^10^8
x^2 + 6x = 9765625x + 5159780352

Rearrange the equation:

x^2 + 6x - 9765625x - 5159780352 = 0
x^2 - 9765619x - 5159780352 = 0

Now we can solve this quadratic equation for x using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = -9765619, and c = -5159780352

Plugging in the values:

x = (9765619 ± sqrt((-9765619)^2 - 4*1*(-5159780352))) / 2*1

x = (9765619 ± sqrt(95321440924641 + 20639121408)) / 2

x = (9765619 ± sqrt(95321461563799)) / 2

x ≈ (9765619 ± 9759995) / 2

x ≈ 9765625 or x ≈ -23

Therefore, the solutions for x are approximately x = 9765625 or x = -23.