Yes of course. I did not take my own advice and use a calculator. I was babysitting my grandson at the time and probably got distracted. Apologies. The arithmetic gives 241.8, which translates into 20 years, 1 month, and about 24 days (depending on what number we use for days per month).
That answer is, however, unrealistic. We are talking about semi-annual compounding. The 0.3 in 40.3 is inconsistent with the terms of the problem as you originally gave it: the number of compounding periods must be an integer. Jonah addressed this issue in at least one of his posts.
How interest is treated between compounding periods is a matter of statute, contract, and commercial custom rather than mathematics. Your problem does not say what to do with fractional periods. There is no uniform practice. The bank I was associated with for over forty years changed its method when I became in charge of deposit pricing. Before then, when we compounded “continuously,” we ignored days: no interest was paid for days since the end of the preceding month. After then, when we compounded monthly, we paid simple interest for days since the end of the preceding month. And no one that I am aware of pays interest for fractions of days, let alone fractions of minutes. Unless your problem specifies how interest is to be computed for fractional periods, it is unanswerable.
Moreover, it may be impossible to give an exact numeric answer to the question. Once we are into logs we have left the realm of rational numbers and what is exactly expressible in a finite number of figures. A carefully worded question will avoid this issue: “How many periods will elapse before the value of principal has increased four times“ is not asking when it is exactly four times, which may be at no time that can be expressed exactly. Students need to look very carefully at the wording of problems.
Finally, we get to the issue of no calculators in examinations. This means we must give answers that avoid complex computations. My preferred answer to the question of “how many compounding periods will elapse BEFORE principal has increased at least fourfold given annual interest at 7% compounded semiannually“ would be
[MATH]n = \left \lceil \dfrac{log(4)}{log(1.035)}\right \rceil - 1.[/MATH]
If the notation is strange, look up ceiling function. Notice that no approximation sign was used; the answer is an integer.
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