$415,000 compounded by 5% in how many years does it double

The amount doesn't matter....we have

2P = P(1.05)^t    divide both sides by P ... (see why the amount doesn't mattter??)

2 = (1.05)^t       take the log of both sides

log 2 = log(1.05)^t     and by a log property, we have

log 2 = t log(1.05)    divide both sides by  log(1.05)

log 2 /  log(1.05) = t = about 14.2 years