Today's soundtrack is blink-182: Takeoff Your Pants and Jacket, a classic pop-punk album. I still remember seeing the music video for "The Rock Show" on MTV for the first time and being like, "Whoa! This is rad!"
This evening, I'm learning about different ways that we can put exponential functions to use.
Using logarithms to solve our exponential functions, we can determine how long it would take for a value (p) to double. For example, if we are told that a house cost $400,000 in 1992 and its value was appreciating at a rate of 6.1% per year, we could determine in what year its value would have doubled by doing the following:
Plug our known values into A=P(1+[r/n])ⁿᵗ
800,000 = 400,000(1.061)ᵗ
Divide both sides by P
2 = 1.061ᵗ
Take the logs
log2 = log1.061ᵗ
Apply the power rule
log2 = tlog1.061
Isolate the variable
t=(log2/log1.061)
Use technology to calculate
t=11.71
Now that we know t's value, we know that our original bracketed term (1.061) must be raised to the power 11.71 to equal 2. Thus, 11.71 is the doubling time of this equation.
We can take this a step further and write the original function as an exponential function with a base of 2 to represent doubling time. 2 must be raised to the power of 1 to double the value; we represent its power as t/[doubling time] to show that when the numerator and denominator are the same value, P is doubled. If we were to do this with the house in the example above, our equation would be A = 400,000(2)^(t/11.71). So if t = 11.71, P = 800,000.
So "doubling time" is the time (or distance, etc.) that it takes for a value to double. That concept is related to exponential growth. On the flip side of the coin is exponential decay and "half life," the measure (of time or distance, etc.) that it takes for a value to halve.
Much like the concept presented above, we can rewrite an equation of exponential decay to represent a value's half-life. An example would be 50=100(0.96)¹⁶⋅⁹⁸, which can be rewritten as 50=100(1/2)⁽¹⁶⋅⁹⁸/¹⁶⋅⁹⁸⁾.
In either case, whether we are being asked to rewrite an equation with its doubled time or its half-life, we are doing a two-step process: first, we must solve the original equation and find the value that will go into the exponent position (t), and then we can rewrite the equation using that information. Once we've determined t, a doubling time equation will use the base 2; a half-life equation will use the base 1/2.
Important note: In the exponent t/n, the doubling time or half-life time will go into position n (the denominator); the unknown time will go into t (the numerator).
Another important note when choosing an equation:
If compounding occurs more frequently than measured time
If compounding occurs more frequently than the measured period of time, we use the equation A = P(1+[r/n])ⁿᵗ, with n representing the number of times that the compounding occurs in t.
If compounding occurs less frequently than measured time
If compounding occurs less frequently than the measured periods of time, we use the equation A = P(1+r)⁽ᵗ/ⁿ⁾, with n representing the number of times that t occurs before our rate compounds.
We now move on to learning something new. We already know that common logarithms have base 10. But there's another kind that we'll start working with now: natural logarithms, which have base e, a value related to growth. The e value can be accessed on the TI-83 series of calculators by pressing [2nd], [÷].
That's all for today. Next time, I'll be learning more ways that exponential functions can be applied to real-life scenarios!