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.

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

step-by-step explanation:

i hate black

(x3 + 6x2 + 2x - 6) • (x - 1)

step-by-step explanation:

polynomial long division :

3.2     polynomial long division

dividing :   x4+5x3-4x2-8x+6

                              ("dividend")

by         :     x-1     ("divisor")

dividend     x4   +   5x3   -   4x2   -   8x   +   6

- divisor   * x3     x4   -   x3            

remainder         6x3   -   4x2   -   8x   +   6

- divisor   * 6x2         6x3   -   6x2        

remainder             2x2   -   8x   +   6

- divisor   * 2x1             2x2   -   2x    

remainder               -   6x   +   6

- divisor   * -6x0               -   6x   +   6

remainder                     0

quotient :   x3+6x2+2x-6   remainder:   0