Fundamental Math
IT100 Session 7: Exponent and Logarithmic systems
5 Sept 2017
Agenda
Agenda
- Exponential Functions
- Logarithmic Functions
- Graphs of logarithmic Functions
- Exponential and logarithmic Models
- Fitting exponential Models to Data
Exponential Functions (4.1)
Binary Growth rates
\[ \begin{array}{rccc} 0&2^0&1&o\\ 1&2^1&2&oo\\ 2&2^2&4&oooo\\ 3&2^3&8&oooooooo\\ 4&2^4&16&oooooooooooooooo\\ 5&2^5&32&oooooooooooooooooooooooooooooooo\\ 6&2^5&64&oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo\\ \end{array} \]
Exponential Growth
| Rate | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| \( \large 1^n \) | 1 | 1 | 1 | 1 | 1 | 1 |
| \( \large 2^n \) | 2 | 4 | 8 | 16 | 32 | 64 |
| \( \large 4^n \) | 3 | 9 | 27 | 81 | 243 | 729 |
| \( \large 6^n \) | 6 | 36 | 216 | 1296 | 7776 | 46656 |
| \( \large 8^n \) | 8 | 64 | 512 | 4096 | 32768 | 262144 |
| \( \large 10^n \) | 10 | 100 | 1000 | 10000 | 100000 | 1000000 |
| \( \large 12^n \) | 12 | 144 | 1728 | 20736 | 248832 | 2985984 |
Exponential Growth
Interest Rate Growth
| Rate | 1yr | 2yr | 3yr | 4yr | 5yr | 6yr | 7yr | 8yr |
|---|---|---|---|---|---|---|---|---|
| 1% | 1.01 | 1.02 | 1.03 | 1.04 | 1.05 | 1.06 | 1.07 | 1.08 |
| 5% | 1.05 | 1.10 | 1.16 | 1.22 | 1.28 | 1.34 | 1.41 | 1.48 |
| 10% | 1.10 | 1.21 | 1.33 | 1.46 | 1.61 | 1.77 | 1.95 | 2.14 |
| 15% | 1.15 | 1.32 | 1.52 | 1.75 | 2.01 | 2.31 | 2.66 | 3.06 |
| 20% | 1.20 | 1.44 | 1.73 | 2.07 | 2.49 | 2.98 | 3.58 | 4.30 |
| 25% | 1.25 | 1.56 | 1.95 | 2.44 | 3.05 | 3.81 | 4.77 | 5.96 |
| 30% | 1.30 | 1.69 | 2.20 | 2.86 | 3.71 | 4.83 | 6.27 | 8.16 |
Interest Rate Growth
Degradation Rates
| Rate | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| ½ | 50.0000 | 25.0000 | 12.5000 | 6.2500 | 3.1250 | 1.5625 |
| ¼ | 25.0000 | 6.2500 | 1.5625 | 0.3906 | 0.0977 | 0.0244 |
| 1/6 | 16.6667 | 2.7778 | 0.4630 | 0.0772 | 0.0129 | 0.0021 |
| 1/8 | 12.5000 | 1.5625 | 0.1953 | 0.0244 | 0.0031 | 0.0004 |
| 1/10 | 10.0000 | 1.0000 | 0.1000 | 0.0100 | 0.0010 | 0.0001 |
| 1/12 | 8.3333 | 0.6944 | 0.0579 | 0.0048 | 0.0004 | 0.0000 |
Degradation Rates
Degradation as percentage
| Rate | 1yr | 2yr | 3yr | 4yr | 5yr | 6yr | 7yr | 8yr |
|---|---|---|---|---|---|---|---|---|
| -1% | 99.0 | 98.0 | 97.0 | 96.1 | 95.1 | 94.1 | 93.2 | 92.3 |
| -5% | 95.0 | 90.3 | 85.7 | 81.5 | 77.4 | 73.5 | 69.8 | 66.3 |
| -10% | 90.0 | 81.0 | 72.9 | 65.6 | 59.0 | 53.1 | 47.8 | 43.0 |
| -15% | 85.0 | 72.3 | 61.4 | 52.2 | 44.4 | 37.7 | 32.1 | 27.2 |
| -20% | 80.0 | 64.0 | 51.2 | 41.0 | 32.8 | 26.2 | 21.0 | 16.8 |
| -25% | 75.0 | 56.3 | 42.2 | 31.6 | 23.7 | 17.8 | 13.3 | 10.0 |
| -30% | 70.0 | 49.0 | 34.3 | 24.0 | 16.8 | 11.8 | 8.2 | 5.8 |
Degradation as percentage
Exponent Rules
\[ \Large\begin{array}{rc} \hbox{Product Rule} & {a^m\times a^n} = a^{m+n} \\ \hbox{Quotient Rule} & {a^m\over a^n} = a^{m-n} \\ \hbox{Product Rule} & (a^m)^n = a^{m \times n} \\ \hbox{Zero Exponent Rule} & a^0 = 1 \\ \hbox{Negative Exponent Rule} & a^{-n} = {1\over a^n} \\ \hbox{Power of ratio} & \left({a\over b}\right)^n = {a^n \over b^n}\\ \end{array} \]
Population Growth
\[ \Large N(t) = P \ b^t \]
- \( \Large N \) : Population at time \( \Large t \)
- \( \Large P \) : Initial population
- \( \Large b \) : base of population growth
- \( \Large t \) : Time in years
*Population of 3 Provinces *
| Provinces | 1999 | 2014 | 2016 |
|---|---|---|---|
| Chiang Mai | 1,587,465 | 1,678,284 | 1,735,762 |
| Songkla | 1,223,833 | 1,401,303 | 1,417,440 |
| Khon Khan | 1,747,730 | 1,790,049 | 1,801,753 |
| Phuket | 241,489 | 378,364 | 386,605 |
Calculations
\[ \large\begin{array}{rclcrclrl} 1.736 &=& 1.587\ b_{cm}^{17}& &1.678 &=& 1.587\ b_{cm}^{15} \\ 1.093888 &=& b_{cm}^{17}& & 1.057341 &= & b_{cm}^{15}\\ 1.00529&=&b_{cm} & & 1.00372 &=& b_{cm} &b_{cm} & =1.004505 \\ \\ 1.417 &=& 1.224 b_{sng}^{17} & & 1.401 &=& 1.224 b_{sng}^{15}\\ 1.15768 &=& b_{sng}^{17} & & 1.14461 &=& b_{sng}^{15}\\ 1.00865 &=& b_{sng} & & 1.009045 &=& b_{sng}&b_{sng}&=1.00847\\ \\ 1.801 &=& 1.747\ b_{kk}^{17} & & 1.790 &=& 1.747\ b_{kk}^{15}\\ 1.03091 &=& b_{kk}^{17} & & 1.024614 &=& b_{kk}^{15}\\ 1.001792 &=& b_{kk} & & 1.001622 &=& b_{kk}&b_{kk} &= 1.001707 \\ \\ 0.387 &=& 0.241\ b_{pk}^{17} & & 0.378 &=& 0.241\ b_{pk}^{15}\\ 1.605809 &=& b_{pk}^{17} & & 1.568465 &=& b_{pk}^{15}\\ 1.028252 &=& b_{pk} & & 1.030461 &=& b_{pk}&b_{pk} &= 1.029357 \\ \end{array} \]
Population Modelling
Challenge
Device on Internet
In 2005 there were 5 million devices connected to the Internet in Thailand. In 2017 the number of devices has grown to 35 million.
Determine the average annual growth rate assuming linear growth.
Determine the annual rate of growth assuming constant exponential grown.
Estimate the number of devices connected in the year 2020?
Compound Interest
\[ \Large A(t) = P\left(1+{r\over n}\right)^{nt} \]
- \( \Large P \) : Principle
- \( \Large t \) : Time in years
- \( \Large r \) : Annual Percentage Rate (APR)
- \( \Large n \) : Compound periods per year
Logarithmic Functions (4.3)
Logarithmic conversions
\[ \Huge\begin{array}{rcl} y &=& \log_{10}(x)\\ x &=& 10^y\\ x_1 \times x_2 &=& 10^{\left(log(x_1)+log(x_2)\right)}\\ \sqrt[4]{x} &=& 10^{{\log_{10}(x)\over 4}}\\ \end{array} \]
Log Scale
Slide rule
Slide rule
- Based on corresponding log10 of the C/D scales
- Adding L values equals the product of numbers
- Multiplication of L value by 2 used for \( \large x^2 \)
- Division of L value by 2 used for \( \large \sqrt{x} \)
Challenge
Find the following values using the virtual slide rule:
\[ \Large \begin{array}{lrclr} Area of circle: & A & = & \pi r^2 = \pi \times 3^2 & ......(1)\\ Ratios: & {a\over b} &=& {2\over 5} = {?\over 40} = {?\over 10} = {20\over ?}& ......(2)\\ Volume of a box: & h \times l \times 2 &=& 5 \times 15 \times 20&.....(3)\\ Volume of a cylinder:& \pi r^2 h &=& \pi 4^2 \times 10 & ......(4)\\ \end{array} \]
Log-Linear and Log-log plots
Graphs (4.2, 4.4)
Linear - Linear Plot
\[ \Large \begin{array}{rcl} y &=& 3x\\ y &=& 25/x\\ y &=& x^2\\ y &=& x^3\\ y&=& 2^x\\ y&=& 3^x\\ \end{array} \]
Log Plots
Challenge
Note the scale of the linear-linear,log-linear and log-log plots
Determine which relationships become linear in log-linear plots
Determine which relationships become linear in log-log plots
Note the angles and shape of the curves in each of the plots
Exponential and Logarithmic Models (4.7)
Challenge
- Plot the data for Population A and B
- Determine the best trendline
- Determine the R2 for the trendline
- Identify which population is moving toward steady-state