quiz8solutions

# quiz8solutions - MAC1147 Quiz#8 In the top-right corner of...

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Unformatted text preview: MAC1147: Quiz #8 10/27/2009 In the top-right corner of a clean sheet of paper, write your name, UFID, and section number. Please use a pen with blue or black ink. When you are nished, FOLD your paper in half lengthwise and write your name on the back. 1. Consider the equation: 2 ln(x) + ln(x + 3) = ln(4x) (a) State the one-to-one property for logarithmic functions: If loga x = loga y , then x = y (b) Finish the statement of the following property: n · loga u =loga (un ) (c) Finish the statement of the following property: loga u + loga v = loga (uv ) (d) Use parts (a), (b), and (c) to solve the equation above. 2 ln(x) + ln(x + 3) = ln(4x) ⇒ ln(x2 (x + 3)) = ln(4x) ⇒ x3 + 3x2 + 4x = 0 ⇒ x(x + 3)(x + 1) = 0 ⇒ x = 0; x = −3; x = −1. But x = −3 is not in the domain, so only x = 0 and x = −1 are solutions. 2. Let f (x) = ln(3x − 4), and let g (x) = e2x + 1 (a) Find the domain of (f ◦ g )(x) (f ◦ g )(x) = f (g (x)) = f (e2x +1) = ln(3(e2x +1) − 4) = ln(3e2x − 1). The domain of ln(x) is (0, ∞), so we must set 3e2x − 1 > 0 ⇒ e2x > ln( 1 ) ln( 1 ) 1 ⇒ x > 23 , or ( 23 , ∞) in interval notation. 3 (b) Find g −1 (x) x = e2y + 1 ⇒ e2y = x − 1 ⇒ ln(x − 1) = 2y ⇒ g −1 (x) = 1 ln(x−1) 2 ...
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