How do I find the range of these logarithmic functions? ln ⁡ ( 3 x 2 − 4 x + 5 ) , log 3 ⁡ ( 5 + 4 x − x 2 ) .

Question

How do I find the range of these logarithmic functions?                          ln        ⁡        (        3                  x          2                −        4        x        +        5        )        ,                                            log          3                ⁡        (        5        +        4        x        −                  x          2                )        .

SoroAlcommai9 · Accepted Answer

You can only take a logarithm of a number greater than zero.So you need   3      x    2    −  4  x  +  5  &amp;gt;  0 in the first case. Completing the square give you             (      x      −              2        3            )        2    +      11    9  . We see that the quadratic is always greater than       11    9   and goes to infinity. Therefore the range is   [  ln  ⁡      (          11      9        )    ,  →  ⟩For the second one, you want   −      x    2    +  4  x  +  5  &amp;gt;  0. We first solve   −      x    2    +  4  x  +  5  =  0. This gives x=−1 or x=5 as you found. Because the coefficient for       x    2   is negative, this means that the quadratic is positive when −1&amp;lt;x&amp;lt;5. The maximum is attained at   x  =  −      b          2      a        =  2, whith a value of 9.So we can make the argument of the log very close to zero but never greater than 9. As       log    3    ⁡  (  9  )  =  2, the range is   ⟨  ←  ,  2  ]

How do I find the range of these logarithmic functions? ln(3x^2−4x+5), log_3(5+4x−x^2). how should I approach questions like this ?

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