What is the value of (-1)^{r n}C_r (1+r ln 10)/ (1+ ln (10ⁿ))^r where n is a positivie integer???

do you mean the summation of this series ??

(-1)^r nC_r (1+r ln 10)/(1+ ln (10ⁿ))^r

Denominator is constant. take it equal to k.

k = 1+ ln (10ⁿ)

(-1)^r nC_r (1+r ln 10)/k^r

= (-1)^r nC_r*1/k^r  + ln10 *(-1)^r nC_r r/k^r                ..............(i)

Put summation.

Summation of (-1)^r nC_r*1/k^r from r=0 to n = (1-1/k)ⁿ

Summation of (-1)^r nC_r r/k^r = -n/k * (1-1/k)^n-1

[This can be evaluated by expanding (1-x)ⁿ then differentiating both sides wrt x, then multiplying both sides by x and putting x=1/k]

Putting the above values in equation (i),

(1-1/k)ⁿ + ln10 * (-n/k * (1-1/k)^n-1)

= (n.ln10)ⁿ/kⁿ - n.ln10.(n.ln 10)^n-1/kⁿ

= 0