How do I solve this? without logarithm

Question

A fully loaded Toyota Forerunner sells for $42,000. It depreciates 12% per year from the date of

purchase. How many years will it take to be worth half its value?

I think part of the equation is 42,000(.88)^x

idk what it means for half, half of 42,000 is 21000

how would I solve this without a calculator, I forgot how to...

Answer 1

You are right about the right side of the equation. The left side of the equation is the value at the end of depreciation, which as you said is half of 42,000, so 21,000. So our full starting equation is:

21,000 = 42,000(.88)^x

We can divide both sides by 42,000, and we know that 21k is half of 42k:

0.5 = (0.88)^x

Next, we'll take the natural log (ln) of both sides, and then with the rules of logarithms we know that we can bring the exponent of the argument of a log down in front of the log:

ln(0.5) = ln(0.88)^x

ln(0.5) = xln(0.88)

And then we get x by itself:

ln(0.5) / ln(0.88) = x = 5.42 years

You said that you can't use a calculator, so you would just circle ln(0.5) / ln(0.88) as your answer; there's nothing you can do from there without a calculator.