Today's soundtrack is Steve Vai: Passion and Warfare, an album that really showcases this guitar master's artistic virtuosity.
This afternoon, I'm learning more ways that we can apply exponential functions.
We learned yesterday about doubling time and half-life. These concepts can be expanded to include tripling, quartering, increasing tenfold, etc.
We use the equation A = P(C)ᵗ/ⁿ.
A = final amount
P = initial amount
C = a constant (tenfold, tripling, halving, quartering, doubling, etc.)
t = time
n = the period of time that it takes for C to apply
The Richter Scale
We'll now discuss the Richter Scale, a scale used to measure the intensity of earthquakes. The scale runs from 1-10. A single higher magnitude is ten times as intense as the magnitude before it - thus, a 6.0 magnitude earthquake is 10x as powerful as a magnitude 5.0 earthquake is. Following this same thread, a 7.0 earthquake is 100x as powerful as is a 5.0 magnitude earthquake, for it is 10x10 as intense. So we can see that the Richter Scale is an exponential scale, meaning that we can compare different magnitudes by powers. Therefore, instead of saying that an 8.0 magnitude earthquake is -10x as powerful as a magnitude 9.0 earthquake, we could say that a magnitude 8.0 earthquake is 10⁻¹ as powerful as a 9.0 magnitude earthquake, and a 9.0 earthquake is 10³ as powerful as is a 6.0 magnitude earthquake.
When comparing earthquakes whose orders of magnitude are not exactly 1 order apart (for example, a 6.5 earthquake and an 8.1 earthquake), we apply each value of magnitude as an exponent of the intensity between each point, then we divide one by the other. For example, if we wanted to compare an earthquake of magnitude 8.1 to an earthquake of magnitude 6.5, we would calculate 10⁸⋅¹/10⁶⋅⁵, which is 39.81; thus, we know that a magnitude 8.1 earthquake is 39.8 times as powerful as a magnitude 6.5 earthquake.
The Decibel Scale
The Decibel Scale measures sound in decibels (dB). The scale runs from 0-140. Every ten points is an increase of tenfold; thus, an increase of twenty points is an increase of intensity of 10², or 100.
The pH Scale
The pH scale measures acidity of solutions. The scale runs from 0-14. It is logarithmic; each point "corresponds to a ten-fold change in H⁺ ion concentration" (source). The higher up the scale a solution is, the more alkaline it is; the lower down the scale it is, the more acidic it is.
That's all for today! Next time, we'll start work on graphs of exponential functions!