Find all values of x that solve the equations log8 x 2 lnx

Find all values of x that solve the equations. log_8 (x) = 2 ln(x) + ln(x - 5) = ln(12 - x)

Solution

7) log₈(x) = 2

=> x = 8² = 64

8) ln(x) + ln(x-5) = ln(12-x)

ln(x(x-5)) = ln(12-x)

=> x² - 5x = 12 - x

=> x² - 4x - 12 = 0

=> (x-6)(x+2) = 0

=> x = 6 or x = -2

but x can not be -2 because then ln(x-5) is not defined.

=> x = 6