We need to figure some things out first. We know the initial value of our vehicle is $18,500. We want to know how long it will take for it to depreciate to $5,000, given a 10.75% annual depreciation rate. Since every year the car depreciates by 10.75%, the value of the car for a given year must be 1-0.1075=0.8925. The value for one year must be 89.25% of the value of the previous year.
So, now that we have this information, let's create an equation, $18,500*0.8925^t=$5,000. t=number of years. We need to solve for t.
Let's first divide by 18,500 on both sides. So, we get 0.8925^t=5,000/18,500. 0.8925^t=0.27.
Now, we need to take the natural log on both sides. So, ln 0.8925^t=ln 0.27. In the case of natural logs, if there is an exponent, we can bring it down. So t*ln 0.8925=ln 0.27. Divide ln 0.8925 from both sides. So t=ln 0.27/ln 0.8925.
t=11.5 years, or 11 years and a half