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

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