MHB (ask) Calculating Logarithmic Question

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Calculating the expression log_2(log_3(65)) + 3*log_5(log_5(9) is challenging without a calculator, as it involves complex logarithmic functions. Despite attempts to simplify using the change of base formula, users found it complicated and unable to yield the expected answer of 3. The discussion highlights that calculators are essential for evaluating such logarithmic expressions accurately. Some users noted discrepancies in answers, with one mentioning a calculator result of 2.506, likely due to rounding. Ultimately, the consensus is that for complicated logarithmic calculations, using a calculator is advisable.
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May I know how to calculate this question without a calculator? The final answer of this question is 3 but I really have no idea how to work on it to get the final answer.

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Jdtbtb_sp said:
May I know how to calculate this question without a calculator? The final answer of this question is 3 but I really have no idea how to work on it to get the final answer.

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Is it something to do with the changing base formula? I have tried using the changing base formula, hoping through this I could simplify it but it went even complicated. Then I was stuck, don't know how to solve it...
 
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Even if you change all the logarithms to the same base there is not going to be any simple way to calculate that "log(log)" without a calculator. That's why calculators were invented!
 
Country Boy said:
Even if you change all the logarithms to the same base there is not going to be any simple way to calculate that "log(log)" without a calculator. That's why calculators were invented!

Thank you for replying. Next time I should use a calculator for such a "log(log)" question. Thank you for spending time to answer my post.
 

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