independanteng

independanteng

Answered

2022-10-13

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

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Answer & Explanation

SoroAlcommai9

SoroAlcommai9

Expert

2022-10-14Added 13 answers

You can only take a logarithm of a number greater than zero.
So you need 3 x 2 4 x + 5 > 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 > 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<x<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 ]

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