9514 1404 393
Answer:
x = 2
Step-by-step explanation:
[tex]\log_2(x+14)=2+\log_{\sqrt{2}}(x)\qquad\text{given}\\\\\dfrac{\log(x+14)}{\log(2)}=2+\dfrac{\log(x)}{\frac{1}{2}\log(2)}\qquad\text{change of base formula}\\\\\log(x+14)=2\log(2)+2\log(x)\qquad\text{multiply by $\log(2)$}\\\\log(x+14)=\log((2x)^2)\qquad\text{simplify}\\\\x+14=4x^2\qquad\text{take antilogs}\\\\(x-2)(4x+7)=0\qquad\text{rearrange and factor}\\\\\boxed{x=2}\qquad\text{x=-7/4 is an extraneous solution}[/tex]
The solution to the quadratic is the (positive) value of x that makes a factor zero.
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The relevant rules of logarithms are ...
log(ab) = log(a) +log(b)
log(a^b) = b·log(a)
logₙ(a) = log(a)/log(n)
