P = (r * PV) / (1 - (1 + r)**-n)

P * (1 - (1 + r)**-n) = (r * PV)

1 - (1 + r)**-n = (r * PV) / P

1  = (r * PV / P) + (1 + r) ** -n

1  - (r * PV / P) = (1 + r) ** -n

1  - (r * PV / P) = 1 /(1 + r) ** n

((1 + r) ** n) * (1  - (r * PV / P)) = 1

(1 + r) ** n = 1 / (1  - (r * PV / P)) # take the log of both sides: http://mathonweb.com/help_ebook/html/expolog_4.htm

n * log(1 + r) = log(1 / (1  - (r * PV / P))) # divide both sides by log(1 + r)

n  = log(1 / (1  - (r * PV / P))) / log(1 + r) # finally done!

from numpy import log10

def get_periods_from_payment(present_value, rate_per_period, payment):
    rate_as_decimal = rate_per_period / 100
    return log10(1 / (1  - (rate_per_period * present_value / payment))) / log10(1 + rate_per_period)

# test get_periods_from_payment()
result = get_periods_from_payment(35000, 0.5, 295.35)
print(result)
assert result == 180