Solve the following equations.

ln (x + 2) + ln (x - 2) = ln (9x - 24)

x = 4 and, x = 5

Step-by-step explanation:

Data provided:

ln (x + 2) + ln (x - 2) = ln (9x - 24) ... (1)

Now,

from the properties of natural log, we have

ln (A) + ln (B) = ln (AB)

applying the above property in the equation given, we get

ln (x + 2) + ln (x - 2) = ln ((x + 2) (x - 2))

or

ln (x + 2) + ln (x - 2) = ln (x² - 2²)

on substituting the above result in the equation (1)

ln (x² - 2²) = ln (9x - 24)

taking the anti-log both sides, we get

(x² - 2²) = (9x - 24)

or

x² - 4 = 9x - 24

or

x² - 4 - 9x + 24 = 0

or

x² - 9x + 20 = 0

or

x² - 4x - 5x + 20 = 0

or

x (x - 4) - 5 (x - 4) = 0

or

(x - 4) (x - 5) = 0

thus,

x = 4 and, x = 5