Find the value of b for which 1+e^b+e^2b+e^3b + ... = 9

This is a geometric series, let r=e^b then we have

1+r+r^2+r^3 + ... = 9

1 / (1-r) = 9

1-r = (1/9)

r=1 - (1/9)

r=8/9

e^b=8/9 take ln of both sides

ln (e^b) = 8/9 ln (x^y) = y*ln (x) and ln (e) = 1 so

b*ln (e) = b=ln (8/9)

thus

b=ln (8/9)