In the equation given, P(t) represents the price of the batteries. We want to figure out what t is equal to when P(t) is 4. In order to do that, we can set up the following equation:
We need to figure out how to isolate t. we could start by dividing both sides by 1.1 to isolate the term that contains e
Now, we need to isolate t. We can do this by taking the natural log of both sides (the natural log is just a special logarithmic function in which the base is e):
Using our log rules, we can bring 0.047t to the front because it is a power:
because ln is the same thing as log base e, we know that ln(e) has to be equal to 1 (you can think about it this way: e^1 is e, which means ln(e) is 1). Therefore, we can simplify it to get the following equation:
Now, we can use a calculator to solve for t:
We are looking for the year when the price is 4 dollars, and we have figured out that the year will be 27 years after 1980. In other words, it will be 1980 + 27, or 2007
Therefore, the price will reach $4 in 2007