Alan

So, I guess you are looking for some function called halflog_a, say, defined in such a way that halflog_a(halflog_a(b))= log_a(b).

I don’t know of any such (though that doesn’t necessarily mean it doesn’t exist!)

This was answered here: https://web2.0calc.com/questions/what-is-the-value-of-p-q, where I interpreted the RHS of your equation as (2x - 1)*2.  If the RHS is meant to be (2x - 1)^2 you should be able to use the same technique to complete the problem.

Oops! I read it as 1*2*3*...*97*98*99.   It isn't!  

With z, say, as the product of the odd integers from 1 to 99 we have:  

z→3^26*5^12*7^8*11^5*13^4*17^3*19^3*23^2*29^2*31^2*37*41*43*47*53*59*61*67*71*73*79*83*89*97

so n is 26

(and, yes, this is known as a double factorial.  Thanks for pointing out my error Guest.).

99!→2^95*3^48*5^22*7^16*11^9*13^7*17^5*19^5*23^4*29^3*31^3*37^2*41^2*43^2*47^2*53*59*61*67*71*73*79*83*89*97

n is 48.

99! is 99*98*97*...*3*2*1    It is a single factorial, not a double factorial.

When x = 8 we have (x-8) = 0

For any other value of x we have (x-8) is either positive or negative. If you square a positive number you get a positive number; if you square a negative number you get a positive number. 

When k = 8 we have \(f(x) = x^2-16x+64+1 \text{ or } f(x) = (x - 8)^2 + 1\)

No matter what the value of x is, (x-8)² is always zero or positive, Hence f(x) is always greater than zero.

\( \text{In }\triangle ABC, \text{ points X and Z are on }\overline{AB} \text{and Y is on } \overline{AC} \text{such that }\overline{XY}\parallel \overline{BC} \text{ and } \overline{ZY}\parallel \overline{XC}.\\\text{ If AZ = 8 and ZX = 4, then what is XB?}\)

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Try a few terms! 

a1=1/(1-x), a2=1/(1-a1)→1/(1-1/(1-x))→1-1/x, a3=1/(1-a2)→1/(1-(1-1/x))→x,

continuing in this fashion we find

a4=1/(1-x), a5=1-1/x, a6=x ...etc.

So a107=...

(I’m sure you can take it from here)

I think this should probably read as:

\(\text{if } a_1=\frac{1}{1-x},a_2=\frac{1}{1-a_1}, \text{ and }a_n=\frac{1}{1-a_{n-1}}\text{, for }n\ge2, x\ne0,\text{ find }a_{107}\text{ in simplest form}\)

My attempt at a solution:

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