Solve this ln problem using best method lnx 8 lnx 1 2 ln

Solve this ln problem using best method: ln(x + 8) + ln(x – 1) = 2 ln x

Solution

Given ln(x + 8) + ln(x – 1) = 2 ln x

ln is the natural logorithm and it is logarithm with a base \'e\'.

Solution:

ln(x + 8) + ln(x – 1) = 2 ln x

ln(x + 8) + ln(x – 1) - 2 ln x = 0

ln((x + 8)(x – 1)) - 2 ln x = 0 (because ln(a.b) = ln(a) + ln(b) )

ln((x + 8)(x – 1)) - ln x² = 0 (ln(x^a) = a.ln(x))   

ln((x + 8)(x – 1)/x²) = 0   ( ln(a/b) = ln(a) - ln(b) )

log_e((x + 8)(x – 1)/x²) = 0 (ln(x) = log_ex)

(x + 8)(x – 1)/x² = e⁰ (if log_ex = y then x=e^y)

(x + 8)(x – 1)/x² = 1 (a⁰ = 1 a- any number)

(x + 8)(x – 1) = x²

x² +7x-8 = x²

7x-8 = 0

7x = 8

X = 8/7

Therefore the solution for the given equation is x=8/7