##05AB1E, 16 13 bytes##

[Dî2‹#¼žr.n]¾

Explanation

              # implicit input n
[          ]  # infinite loop
 Dî2‹#        # break if n rounded up is less than 2
      ¼       # else, increase counter
       žr.n   # set next n = log(n)
            ¾ # push counter and implicitly print

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05AB1E, 16 13 bytes

[Dî2‹#¼žr.n]¾

Explanation

              # implicit input n
[          ]  # infinite loop
 Dî2‹#        # break if n rounded up is less than 2
      ¼       # else, increase counter
       žr.n   # set next n = log(n)
            ¾ # push counter and implicitly print

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##05AB1E, 16 bytes##

i0ë¹[D1‹#¼žr.n]¾

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##05AB1E, 16 13 bytes##

[Dî2‹#¼žr.n]¾

Explanation

              # implicit input n
[          ]  # infinite loop
 Dî2‹#        # break if n rounded up is less than 2
      ¼       # else, increase counter
       žr.n   # set next n = log(n)
            ¾ # push counter and implicitly print

Try it online