Question

how to solve the logarithmic equation.

Answer 1

Using logarithmic laws: log(a) + log(b) = log(ab)

log₄ (x + 9) + log₄ (x + 21) = log₄ [(x + 9)(x + 21)] = 3

Since we want the equation in terms of x, the next intuitive step is take 4 to the power of both sides, because the left hand side will simplify easily.

$4^{log_4 (x + 9)(x + 21)} = 4^{3}$
$(x + 9)(x + 21) = 64$
$x^{2} + 9x + 21x + 189 = 64$

Solving the quadratic we get:
$x^{2} + 30x + 125 = 0$
$(x + 5)(x + 25) = 0$

$x = -5, -25$

BUT, for the logarithmic equation to be defined, x > -9, so x≠ -25
Thus, the only solution is x = -5.