How do I find the range of these logarithmic functions?
Answer & Explanation
You can only take a logarithm of a number greater than zero.
So you need in the first case. Completing the square give you . We see that the quadratic is always greater than and goes to infinity. Therefore the range is
For the second one, you want . We first solve . This gives x=−1 or x=5 as you found. Because the coefficient for is negative, this means that the quadratic is positive when −1<x<5. The maximum is attained at , whith a value of 9.
So we can make the argument of the log very close to zero but never greater than 9. As , the range is
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