Today's soundtrack is Strapping Young Lad: The New Black, an album that is, quite plainly, FRICKIN HEAVY. There's a reason that SYL's frontman, the legendary and prolific Canadian artist Devin Townsend, is nicknamed "Heavy Devy." Townsend's mastered the "wall of noise;" his sound is both overwhelming and tantalizing at the same time. I love this album.
This evening, I'm learning about exponential functions.
To determine compound interest, we use the equation A = P(1+[r/n])ⁿᵗ. We can use the same formula whether we are working with exponential growth or exponential decay; the only difference is that when working with exponential decay, we will subtract r/n instead of adding it.
Here's how it works:
A is the final amount
P is the starting population
r is the rate of change
n is the number of times the rate is compounded within the timeframe
The exponent t is the timeframe
Let's go through an example.
If I invest $2000 into a high-interest savings account with an interest rate of 4.5% compounded quarterly, how long will it be until my investment doubles?
First, I'm going to substitute my values into the formula: the final amount is going to be 4000, since we are looking for a double of 2000. We know that the rate of change is 4.5%, which in decimal form is 0.045. The rate is compounded four times per timeframe (year). So our equation will be:
4000 = 2000(1+[0.045/4])⁴ᵗ.
There are two ways to solve this equation. The first (and easiest) is inputting it into a graphing calculator, substituting y for 4000, and using the [calc] functions to figure out what we need to know. But this is precalculus 12! We don't take the easy way. We're going to do this algebraically.
First, we divide the whole thing by 2000, giving us 2=(1+[0.045/4])⁴ᵗ. Next, let's simplify the bracketed portion. We get 2=1.01125⁴ᵗ. Now we just have to solve for t. So we take the logs of both sides of the equation and use our powers rule: log2 = 4tlog1.01125. Let's isolate the variable by dividing both sides by 4log1.01125. This gives us t = (log2/4log1.01125). According to my calculator, this equals 15.5 when rounded to the nearest tenth. So it will take us 15.5 years to have our $2000 investment reach $4000.