For what values of b will F(x) = logb x be an increasing function?
2 answers:
In order to answer you need to keep in mind what is a logarithm:
logb x = n is the equivalent of asking "you should raise the base b <span>to <span>what number n</span></span> in order to get x?": bⁿ = x
From this follows that b must be positive, otherwise the function is not defined everywhere in R: for example, take log(-2) x = 0.5 ? This means:
=√(-2) which is impossible in R.
Therefore, both answers A and C are not correct because they take in consideration negative values.
If we take the base between 0 and 1 excluded, we have a fraction whose denominator is bigger than the numerator. Such a fraction, elevated to increasing exponents get smaller, for example:
(1/2)² = 1/4
(1/2)³ = 1/8
(1/2)⁴ = 1/16
Therefore the function is dereasing and answer B is incorrect.
The right answer is, therefore, D) b > 1.
logb x = n is the equivalent of asking "you should raise the base b <span>to <span>what number n</span></span> in order to get x?": bⁿ = x
From this follows that b must be positive, otherwise the function is not defined everywhere in R: for example, take log(-2) x = 0.5 ? This means:
Therefore, both answers A and C are not correct because they take in consideration negative values.
If we take the base between 0 and 1 excluded, we have a fraction whose denominator is bigger than the numerator. Such a fraction, elevated to increasing exponents get smaller, for example:
(1/2)² = 1/4
(1/2)³ = 1/8
(1/2)⁴ = 1/16
Therefore the function is dereasing and answer B is incorrect.
The right answer is, therefore, D) b > 1.
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