How do you solve a logarithm with a non-integer base?How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example: 0.5 x = 8 (where x = − 3 ) log 0.5 ⁡ 8 = − 3How would you solve this, and how would this work for an irrational base (like 2 )?

Question

How do you solve a logarithm with a non-integer base?How would one calculate the log of a number where the base isn&#039;t an integer (in particular, an irrational number)? For example:      0.5    x    =  8       (where     x  =  −  3      )        log          0.5        ⁡  8  =  −  3How would you solve this, and how would this work for an irrational base (like       2  )?

vrtuljakc6 · Accepted Answer

Let&#039;s rewrite this in a different way:      0.5    x    =  8Take the logarithm with respect to any base   a  &amp;gt;  0      log    a    ⁡  (      0.5    x    )  =      log    a    ⁡  8 which becomes  x      log    a    ⁡  0.5  =      log    a    ⁡  8or  x  =                    log        a            ⁡      8                      log        a            ⁡      0.5      You would stop here weren&#039;t from the fact that   8  =      2    3   and   0.5  =      2          −      1       so.  x  =                    log        a            ⁡      8                      log        a            ⁡      0.5        =                    log        a            ⁡              2        3                            log        a            ⁡              2                  −          1                      =            3              log        a            ⁡      2              −              log        a            ⁡      2        =  −  3You need to compute no logarithm, actually.

invioor · Accepted Answer

This is the base changing formula :      log          a        ⁡  (  x  )  =                    log                  b                    ⁡      (      x      )                      log                  b                    ⁡      (      a      )