How would you solve for x in 7 x = 5 x − 4 How would you be able to make these the same base? I tried to put it into a base 10 formula, but it ended up making x = 0

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How would you solve for   x in       7          x        =      5          x      −      4      How would you be able to make these the same base? I tried to put it into a base   10 formula, but it ended up making   x  =  0

vikafa4g · Accepted Answer

Since we only have linear powers of   x (that is, powers of the form   a  x  +  b  , where   a  ,  b constants with   a  ≠  0), then one approach we can take (that puts off logarithms until the very end) is this:      7    x    =      5          x      −      4              7    x    =            5      x              5      4              7    x    =            5      x        625          7    x    ⋅  625  =      5    x      625  =            5      x              7      x          625  =            (              5        7            )        x  At this point, we can take a logarithm (of any base we like) on both sides, and use power rule to isolate   x. More simply, though, we may use the definition of a logarithm. That last equation can be thought of as saying &quot;  x is the exponent on base       5    7   that yields   625.&quot; But this means precisely that  x  =      log                  5        7              ⁡  625.If non-standard bases for logarithms are undesirable for you, then we can apply the change-of-base formula to see that  x  =                                                  log          ⁡          625                                                                  log          ⁡                      5            7                              or  x  =                                                  ln          ⁡          625                                                                  ln          ⁡                      5            7                                ,whichever you prefer. (We can also get by power rule as discussed earlier.)Added: We can do the same kind of thing more generally, too. Suppose that   s  ,  t are positive real numbers,   a  ,  c non-zero real numbers, and   b  ,  d any real numbers. Then the following are equivalent:      s          a      x      +      b        =      t          c      x      +      d              s          a      x            s    b    =      t          c      x            t    d      (      s    a        )    x        s    b    =  (      t    c        )    x        t    d                (              s        a                    )        x                    (              t        c                    )        x              =            t      d              s      b                    (                        s          a                          t          c                    )        x    =            t      d              s      b      At that point, we can use power rule of logarithms or the definition, as described above.

seiyakou2005n1 · Accepted Answer

Take log on both sides (does not matter if you use natural or common logs)  x  log  ⁡  (  7  )  =  (  x  −  4  )  log  ⁡  (  5  )Solving for   x  x  =            −      4      log      ⁡      (      5      )              log      ⁡      (      7      )      −      log      ⁡      (      5      )        ≈  −  19.133Why do I feel like I am doing someone&#039;s homework problem!

How would you solve for x in 7 <mrow class="MJX-TeXAtom-ORD"> x </mrow>

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