solve the logarithmic equation for x. if there are no answers, enter null. round your answer to two decimal…

Explanation:

Step1: Apply log quotient rule

Using (\log a - \log b=\log\frac{a}{b}), rewrite left side: (\log\left(\frac{x + 1}{x + 2}\right)=\log\left(\frac{1}{x}\right))

Step2: Remove logs (since (\log a=\log b\Rightarrow a = b))

Set (\frac{x + 1}{x + 2}=\frac{1}{x}), cross - multiply: (x(x + 1)=x + 2)

Step3: Expand and simplify

Expand left side: (x^{2}+x=x + 2), subtract (x + 2) from both sides: (x^{2}-2 = 0)

Step4: Solve quadratic equation

(x^{2}=2), so (x=\pm\sqrt{2}). But for (\log(x + 1)), (\log(x + 2)), (\log\left(\frac{1}{x}\right)) to be defined:

• (x+1>0\Rightarrow x>- 1)
• (x + 2>0\Rightarrow x>-2)
• (\frac{1}{x}>0\Rightarrow x>0)

So (x=\sqrt{2}\approx1.41) (reject (x =-\sqrt{2}) as it doesn't satisfy (x>0))

Answer:

(1.41)