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16 December, 06:37

expression to approximate log a of x for all positive numbers a, b, and x, where a is not equal to 1 and b is not equal to one

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  1. 16 December, 06:44
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    Question:

    Approximate log base b of x, log_b (x).

    Of course x can't be negative, and b > 1.

    Answer:

    f (x) = (-1/x + 1) / (-1/b + 1)

    Step-by-step explanation:

    log (1) is zero for any base.

    log is strictly increasing.

    log_b (b) = 1

    As x descends to zero, log (x) diverges to - infinity

    Graph of f (x) = (-1/x + 1) / a is reminiscent of log (x), with f (1) = 0.

    Find a such that f (b) = 1

    1 = f (b) = (-1/b + 1) / a

    a = (-1/b + 1)

    Substitute for a:

    f (x) = (-1/x + 1) / (-1/b + 1)

    f (1) = 0

    f (b) = (-1/b + 1) / (-1/b + 1) = 1
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