Question

solve for x.
\log_{5}(x + 7) = -\log_{5}(x + 3) + 1
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.

Explanation:

Step1: Move log term to left

Add $\log_{5}(x + 3)$ to both sides: $\log_{5}(x + 7) + \log_{5}(x + 3) = 1$

Step2: Apply log product rule

Use $\log_a M + \log_a N = \log_a(MN)$: $\log_{5}[(x + 7)(x + 3)] = 1$

Step3: Convert to exponential form

$5^1 = (x + 7)(x + 3)$ → $5 = x^2 + 10x + 21$

Step4: Solve quadratic equation

Rearrange: $x^2 + 10x + 16 = 0$
Factor: $(x + 2)(x + 8) = 0$
Solutions: $x = -2, x = -8$

Step5: Check domain

For $\log_{5}(x + 7)$ and $\log_{5}(x + 3)$, $x + 7 > 0$ and $x + 3 > 0$ → $x > -3$.
$x = -8$: $-8 + 3 = -5 < 0$ (invalid).
$x = -2$: $-2 + 7 = 5 > 0$, $-2 + 3 = 1 > 0$ (valid).

Answer:

-2