Something to the power of a logarithmThis is probably a very obvious question, but here goes...An answer in my textbook claims that 3 log ⁡ n = n log ⁡ 3 and that 4 n 2 ( 3 / 4 ) log ⁡ n = 4 n log ⁡ 3 Why, using more basic laws, is this the case?(Unfortunately Google confuses this question with changing bases, exponentiation being the inverse of log (which is of course related), and similar matters.)

Question

Something to the power of a logarithmThis is probably a very obvious question, but here goes...An answer in my textbook claims that      3          log      ⁡      n        =      n          log      ⁡      3      and that  4      n    2    (  3      /    4      )          log      ⁡      n        =  4      n          log      ⁡      3      Why, using more basic laws, is this the case?(Unfortunately Google confuses this question with changing bases, exponentiation being the inverse of log (which is of course related), and similar matters.)

alomjabpdl0 · Accepted Answer

Recall that   3  =      b                  log        b            ⁡      3      Therefore       3                  log        b            ⁡      n        =            (            b                  log        b            ⁡      3                          )                            log        b            ⁡      n        =      b          (              log        b            ⁡      3      )      (              log        b            ⁡      n      )        =            (            b                  log        b            ⁡      n                          )                            log        b            ⁡      3        =      n                  log        b            ⁡      3

cooloicons62 · Accepted Answer

Hint :   a  =  b implies   log  ⁡  a  =  log  ⁡  b and   log  ⁡      a    b    =  b  log  ⁡  a