Changing base of a logarithm by taking a square root from base?From my homework I found log 9 ⁡ x = log 3 ⁡ x and besides that an explanation that to this was done by taking a square root of the base. I fail to grasp this completely. Should I need to turn log 9 ⁡ x into base 3, I'd do something like log 9 ⁡ x = log 3 ⁡ x log 3 ⁡ 9 = log 3 ⁡ x log 3 ⁡ 3 2 = log 3 ⁡ x 2 but this is a far cry from what I've given as being the correct answer.Substituting some values to x and playing with my calculator I can see that the answer given as correct is correct whereas my attempt fails to yield the correct answer.Now the question is, what are correct steps to derive log 3 ⁡ x from log 9 ⁡ x ? How and why am I allowed to take a square root of the base and the exponent?

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Changing base of a logarithm by taking a square root from base?From my homework I found      log    9    ⁡      x    =      log    3    ⁡            x      and besides that an explanation that to this was done by taking a square root of the base. I fail to grasp this completely. Should I need to turn       log    9    ⁡      x   into base 3, I&#039;d do something like      log    9    ⁡      x    =                    log        3            ⁡              x                            log        3            ⁡              9              =                    log        3            ⁡              x                            log        3            ⁡                        3          2                      =                    log        3            ⁡              x              2  but this is a far cry from what I&#039;ve given as being the correct answer.Substituting some values to x and playing with my calculator I can see that the answer given as correct is correct whereas my attempt fails to yield the correct answer.Now the question is, what are correct steps to derive       log    3    ⁡            x       from       log    9    ⁡      x  ? How and why am I allowed to take a square root of the base and the exponent?

mallol3i · Accepted Answer

You are only one step away!Note that   a  log  ⁡  b  =  log  ⁡      b    a  , so                    log        3            ⁡      x        2    =      1    2        log    3    ⁡  x  =      log    3    ⁡      x          1              /            2        =      log    3    ⁡      x    .

Yesenia Sherman · Accepted Answer

log          9                ⁡        x        =        a                                    ⟺                                                                                      9                  a                                =                x                                                                                  log          3                ⁡                  x                =        b                                    ⟺                          3          b                =                  x                        ⟺                                      (                                    3          b                                                    )                                2                =                              (                                    x                                                    )                                2                        ⟺                          3                      2            b                          =        x                ⟺                                                                                      9                  b                                =                x                                                                       }        →  a  =  b

Changing base of a logarithm by taking a square root from base?From my homework I...

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