Proving log ⁡ x 2 = 2 log ⁡ xHow does log ⁡ x 2 = 2 log ⁡ xCan you do a proof please. I know that this is true but I don't know why.

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Proving   log  ⁡      x    2    =  2  log  ⁡  xHow does   log  ⁡      x    2    =  2  log  ⁡  xCan you do a proof please. I know that this is true but I don&#039;t know why.

scipionhi · Accepted Answer

Apply the exponential to get:            e              log        ⁡        (                  x          2                )              =          x      2        =                  (                  e                      log            ⁡            (            x            )                          )            2        =          e              log        ⁡        (        x        )                    e              log        ⁡        (        x        )              =          e              2        log        ⁡        (        x        )

Averi Mitchell · Accepted Answer

Assuming you have the following understaning of what is a logarithm:  x  =      log          b        ⁡  a    ⟺    a  =      b    x  So, let we have the following values:  x  =      log          b        ⁡  a and   y  =      log          b        ⁡  cWhich means that:  a  =      b    x   and   y  =      log          b        ⁡  cIf we multiply   a and   c:  a  ⋅  c  =      b    x    ⋅      b    y    =      b          x      +      y      Hence, by our definition of logarithm:  x  +  y  =      log          b        ⁡  (  a  ⋅  c  )  =      log          b        ⁡  a  +      log    b    ⁡  c (do you remember that   x and   y are the logarithms of   a and   b?)So, we have prooved that:      log          b        ⁡  (  a  ⋅  c  )  =      log          b        ⁡  a  +      log    b    ⁡  cNow just use that formula in your particular case to get:      log          b        ⁡  (      x    2    )  =      log          b        ⁡  (  x  ⋅  x  )  =  log  ⁡  (  x  )  +  log  ⁡  (  x  )  =  2  log  ⁡  (  x  )

Proving log ⁡ x 2 </msup> = 2 log ⁡<!--

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