I solved this a slightly different way by rearranging and changing the equation to 2^a+(2^a)^2=(2^a)^3 and substituting u for 2^a. I teach calculus, so I usually change equations like this by taking the natural log of both sides. Nice thing about math is that if you do it correctly, two different methods produce the same answer. My answer was (ln(1+sqrt(5))/2)/ln(2), which gave 0.694242, same as your answer. I love looking to see how others solve problems like this! Great challenge.
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I solved this a slightly different way by rearranging and changing the equation to 2^a+(2^a)^2=(2^a)^3 and substituting u for 2^a. I teach calculus, so I usually change equations like this by taking the natural log of both sides. Nice thing about math is that if you do it correctly, two different methods produce the same answer. My answer was (ln(1+sqrt(5))/2)/ln(2), which gave 0.694242, same as your answer. I love looking to see how others solve problems like this! Great challenge.