Question

Math Question is attached. Worth 100 points.

Answer 1

9514 1404 393

Answer:

x = 2

Explanation:

`\log_2(x+14)=2+\log_(√(2))(x)\qquad\text{given}\\\\(\log(x+14))/(\log(2))=2+(\log(x))/((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}`

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)