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Fiesta28 [93]
2 years ago
11

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

Mathematics
1 answer:
amid [387]2 years ago
7 0

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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