I have a vehicle whose original costs is $\$26,000$ and depreciates at an annual rate of $5.5\%$. The person put a down payment of $6000$ on the vehicle and monthly payments sum up to $4800$ per year. The question is, when does the value of the vehicle equal the current expense of the vehicle? So I begin with the equation $$ 26,000(.945)^T = 4,800T+6,000,$$ which then becomes $$ 26(.945)^T = 4.8T+6. $$ I can quickly solve this graphically by putting the left side of the equation into Y1 of my equation editor on the TI-84 and then the right side into Y2; graph, and then find the intersection which has $T$ being approximately $3.26$ years.
My question is - how can I solve for $T$ algebraically? I have carefully looked at the rules of logarithms as have my friends and can't seem to find a way to manipulate the equation so that I can get it into a form $T = \text{some number}$.
Your help would be appreciated.
