How many years would it take for the value of a car purchased at 30 000 to fall to 15 000 if it depreciates by 15% in the first year and 12% every year after that? This is not actually graded homework, but I solved it iteratively (compounding down to 26000 at 15% and so on) and would like to solve it using logarithms to find n as well. The answer is 5 years based on the iterative method at which point it is 14800 or so.
2026-04-03 10:46:41.1775213201
Asset Depreciation - Logarithms
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Let us call $A$ the initial price, $B$ the price after $n$ years, $a$ the depreciation rate for the first year and $b$ the depreciation rate for the following years. When you built your formula, I suppose that you noticed the following relation
$$B = A (1-a) (1-b)^{n-1}$$.
So, going to logarithms is a very good way for finding $n$. For your example $A=30,000$, $B=15,000$,$a=0.15$,$b=0.12$ lead to $n=5.15094$.
It seems that, probably because of some rounding, we do not get exactly the same result