# On the scaling power of logarithmic FCF: #log_(cf) (x;a;b)=log_b (x+a/log_b(x+a/log_b (x+...))), b in (1, oo), x in (0, oo) and a in (0, oo)#. How do you prove that #log_(cf) ( "trillion"; "trillion"; "trillion" )=1.204647904#, nearly?

##### 2 Answers

Calling

with

following with simplifications

finally, computing the value of

We observe also that

This is my continuation to the nice answer by Cesareo. Graphs for ln, choosing b = e and a = 1, might elucidate the nature of this FCF.

#### Explanation:

Graph of

Not bijective for x > 0.

graph{x-2.7183^y+1/y=0 [-10 10 -10 10]}

Graph of y =

Not bijective for x < 0.

graph{-x-2.7183^y+1/y=0 [-10 10 -10 10]}

Combined graph:

graph{(x-2.7183^y+1/y)(-x-2.7183^y+1/y)=0 [-10 10 -10 10]}

The two meet at ( 0, 0.567..). See the graph below. All graphs are

attributed to the power of Socratic graphics facility.

graph{x-2.7128^(-y)+y = 0 [-.05 .05 0.55 .59]}

The answer to the question is 1.02... and Cesareo is right.

See the graphical revelation below.

graph{x-y+1+0.03619ln(1+1/y)=0[-.1 .1 1.01 1.04]}