Japanese | A Nice Olympiads Mathematics Trick 2022 | How to solve for k?

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Input
1^(log_10(0.5 sqrt(5) - 0.5) + (log_10(0.1 e^π) + 1) i) + 10^(log_10(0.5 sqrt(5) - 0.5) + (log_10(0.1 e^π) + 1) i) = 100^(log_10(0.5 sqrt(5) - 0.5) + (log_10(0.1 e^π) + 1) i)
Result
True k=log(0.5sqrt(5)-0.5)+(log(0.1e^pi)+1)i

[@RyanLewis-Johnson-wq6xs]

1^(Log10[0.5+0.5Sqrt[5]])+10^(Log10[0.5+0.5Sqrt[5]])=100^(Log10[0.5+0.5Sqrt[5]]) k= Log10[0.5+0.5Sqrt[5]]