10^n >= 7^(n+1)
ln(10^n) >= ln(7^(n+1))
n*ln(10) >= (n+1)*ln(7)
n*ln(10) >= n*ln(7) + ln(7)
n*ln(10) - n*ln(7) >= ln(7)
n*( ln(10) - ln(7) ) >= ln(7)
n >= ln(7)/( ln(10) - ln(7) )
n >= 5.455696235812878344
n >= 6