Olympiad Maths Prep

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Problem 222

AMC 10/12, early questions
Algebra Difficulty 3.8 Find the answer 19. tekmovanje v znanju matematike za dijake srednjih tehniških in strokovnih šol Državno tekmovanje · Slovenia

Problem:
Kateri izraz je enakovreden izrazu log⁡ab(a−1⋅a3b−1:b−1a23)\log _{a b}\left(a^{-1} \cdot \sqrt{a^{3} b^{-1}}: \sqrt[3]{b^{-1} a^{2}}\right) ?
(A) aba b
(B) ab\frac{a}{b}
(C) 1
(D) −16-\frac{1}{6}
(E) −12-\frac{1}{2}

Official solution

Solution:
Izraz v oklepaju preoblikujemo na skupni korenski eksponent a−1⋅a3b−1:b−1a23=a^{-1} \cdot \sqrt{a^{3} b^{-1}}: \sqrt[3]{b^{-1} a^{2}}= a−66⋅a9b−36:b−2a46\sqrt[6]{a^{-6}} \cdot \sqrt[6]{a^{9} b^{-3}}: \sqrt[6]{b^{-2} a^{4}}. Izraz poenostavimo in damo pod skupni koren: a−1b−16=1ab6=\sqrt[6]{a^{-1} b^{-1}}=\sqrt[6]{\frac{1}{a b}}= (1ab)16=ab−16\left(\frac{1}{a b}\right)^{\frac{1}{6}}=a b^{-\frac{1}{6}}. Logaritem preuredimo v log⁡ab(a−1⋅a3b−1:b−1a23)=log⁡ab(ab)−16=−16\log _{a b}\left(a^{-1} \cdot \sqrt{a^{3} b^{-1}}: \sqrt[3]{b^{-1} a^{2}}\right)=\log _{a b}(a b)^{-\frac{1}{6}}=-\frac{1}{6}.

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