Expected utility maximizer, Applied Statistics

The investor has constant wealth 1 and is o?ered to invest in shares of a project that either gains 3=2 or loses 1 with equal probabilities. Therefore, if the investor obtains shares of this project his wealth is 1+3α = 2 with probability 1/2 and 1 - α with probability 1/2. The investor is an expected utility maximizer with utility index u(z) = ln z. What is the optimal α for this investor? ( α must be between 0 and 1).

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