Solve each of the following equations for x.a) ln ⁡ ( 5 x ) = 0 x = ?b) log 5 ⁡ ( 2 x ) + log 5 ⁡ ( 4 x ) = 2 x = ?

Question

Solve each of the following equations for x.a)   ln  ⁡  (  5  x  )  =  0    x  =  ?b)       log    5    ⁡  (  2  x  )  +      log    5    ⁡  (  4  x  )  =  2    x  =  ?

Nigerkamg5 · Accepted Answer

Suppose   q  =      x    2    +  2  x is rational.Then, by the quadratic formula,   x  =            −      2      ±              4        +        4        q              2    =  −  1  ±      α  , where   α  =  q  +  1 can be any rational number that is not a perfect square (since x is irrational).Now,      x    3    −  6  x  =  (  −  1  ±      α        )    3    −  6  (  −  1  ±      α    )  =  ±      α          3              /            2        −  3  α  ∓  3      α    +  5will be rational if and only if       α          3              /            2        −  3      α   is (since the other terms are always rational). But this is equal to       α    (  α  −  3  ). Since   α  −  3 is rational and       α   irrational, the entire quantity will be rational iff it is zero: that is, iff   α  =  3.Thus the numbers   x that satisfy the condition are   −  1  ±      3

Uroskopieulm · Accepted Answer

b)       log    5    ⁡  (  2  x  )  +      log    5    ⁡  (  4  x  )  =  2        log    5    ⁡  (  2  x  )  (  4  x  )  =  2     take antilog on both sides     8      x    2    =      5    2      8      x    2    =  25        x    2    =      25    8      x  =            25      8          x  =      5          2              2                x  =            5              2              4      x  =            5              2              4    ≈  1.7678

Solve each of the following equations for x. a) ln(5x)=0 x=? b) log_5 (2x)+log_5 (4x)=2 x=?

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