Express logx2 5x 14 logx2 4 as a single logarithm simpli

Express log(x^2 - 5x - 14) - log(x^2 - 4) as a single logarithm, simplify if possible.

Solution

3. log(x^2 -5x -14) -log(x^2 -4)

Use the log property: logA - logB = log(A/B)

log(x^2 -5x -14)/(x^2 -4)

factorise : x^2 -5x -14 = x^2 -7x +2x -14 = x(x-7) +2(x-7)

= (x+2)(x-7)

log(x^2 -5x -14)/(x^2 -4) = log{(x+2)(x-7)/(x+2)(x-2)

= log{(x-7)/(x-2)}

log(x^2 -5x -14) -log(x^2 -4) = log{(x-7)/(x-2)}