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This $$ a\ln q_1+b\ln q_2=q_1^aq_2^b $$ is not true, only $$ a\ln q_1+b\ln q_2=\ln(q_1^aq_2^b) $$ is true. $$ u(\lambda q_1, \lambda q_2)=\lambda u(q_1,q_2) $$ is a sufficient, but not necessary condition for homotheticity of $u$. You can find more on this by searching for questions with the word homothetic on this site. It is indeed true that $$ \lambda^...

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