Question

solve for x.
\log_{2}(x + 2) = 2 - \log_{2}(x + 5)
if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.
\\(x = \square\\)

Explanation:

Step1: Move log term to left

Add $\log_{2}(x + 5)$ to both sides: $\log_{2}(x + 2)+\log_{2}(x + 5)=2$

Step2: Use log product rule

$\log_{a}M+\log_{a}N=\log_{a}(MN)$, so $\log_{2}[(x + 2)(x + 5)] = 2$

Step3: Convert to exponential form

$a^{\log_{a}b}=b$, so $(x + 2)(x + 5)=2^{2}=4$

Step4: Expand and simplify

$x^{2}+7x + 10 = 4\Rightarrow x^{2}+7x + 6 = 0$

Step5: Factor the quadratic

$(x + 1)(x + 6)=0\Rightarrow x=-1$ or $x=-6$

Step6: Check domain

For $\log_{2}(x + 2)$ and $\log_{2}(x + 5)$, $x+2>0\Rightarrow x>-2$ and $x + 5>0\Rightarrow x>-5$. So $x>-2$. $x=-6$ is invalid, $x=-1$ is valid.

Answer:

-1