diff --git a/Grade 10/Math/MCR3U7/Unit 1: Exponential and Logarithmic Functions.md b/Grade 10/Math/MCR3U7/Unit 1: Exponential and Logarithmic Functions.md index 9e3399f..fc35413 100644 --- a/Grade 10/Math/MCR3U7/Unit 1: Exponential and Logarithmic Functions.md +++ b/Grade 10/Math/MCR3U7/Unit 1: Exponential and Logarithmic Functions.md @@ -4,4 +4,94 @@ A function is a relation where each x-value maps to exactly one y-value. -If given a function in the form $`y = af[k(x-d)] + c`$, then let $`(x,y)`$ be the original points, the new points will be $`(\dfrac{1}{k}x+d, ay+c)`$. \ No newline at end of file +If given a function in the form $`y = af[k(x-d)] + c`$, then let $`(x,y)`$ be the original points, the new points will be $`(\dfrac{1}{k}x+d, ay+c)`$. + + +The domain and range of the exponential function is: +- $`D : \{x | x \in \mathbb{R}\}`$ +- $`R : \{y | y > 0, y \in \mathbb{R}\}`$ + +The domain and range of the logarithmic function is +- $`D: \{x | x > 0, x \in \mathbb{R}\}`$ +- $`R: \{y | y \in \mathbb{R}\}`$ + +If $`f(x)`$ is a function, then the inverse is $`f^{-1}(x)`$. The inverse has the following properties: +- Domain of $`f(x)`$ = Range of $`f^{-1}(x)`$ +- Range of $`f(x)`$ = Domain of $`f^{-1}(x)`$ + +Graphically, the inverse of a function is by reflecting the original function over the line $`y=x`$. + +A vertical line test is used to test whether a relation is a function. If any 2 points can be drawn through a vertical line, then that relation is **not a function**. + +To solve/find the inverse of a function, just swap the $`y`$ and $`x`$ and isolate/solve for $`y`$. + +## Exponential Decay/Growth +When the base ($`b`$) is in the range $`0 \lt b \lt 1`$, the exponential funciton is said to have a **exponential decay**, the smaller the base, the stronger the decay. + +When the base ($`b`$) is in the range $`b \gt 1`$, the exponential function is said to have a **exponential growth**, the bigger the base, the stronger the growth. + +## Graphing Exponential Functions +If you have **exponential growth** (meaning your base is greater than $`1`$), use more positive values rather than negative values. + +If you have **exponential decay** (meaning your base is in the range $`(0, 1)`$), use more negative values rather than positive values. + +**Don't forget the asymtote.** + +## Logarithmic Function +The logarithmic function is the **inverse** of the exponential function. + +In essence, if $`x = b^y`$, then $`\log_b x = y`$ + +**Note:** The logarithm is defined only for $`b > 0, b \ne 1`$ + +**Note 2:** The symbol $`ln`$ is $`log_e`$, we usually call it the **natural log**. + +## Logrithm Laws + +1. $`\log_b(b^x) = x`$ +2. $`b^{\log_b(x)} = x`$ +3. $`\log_b(1) = 0`$ +4. $`\log_b(b) = 1`$ + + +|Law|Form|Example| +|:--|:---|:------| +|Change Of Base (COB)|$`\log_a(b) = \dfrac{\log_m(b)}{\log_m(a)}`$|$`\log_2(5) = \dfrac{\log_{10}(5)}{\log_{10}(2)}`$| +|Change Of Base (COB)|$`\log_a(b) = \dfrac{1}{\log_b(a)}`$|$`\log_2(5) = \dfrac{1}{\log_5(2)}`$| +|Power Law|$`\log_b^m(x^n) = \dfrac{n}{m}\log_b(x)`$|$`\log_{2^2}(4^3) = \dfrac{3}{2}\log_2(4)`$| +|Product Law|$`\log_b(xy) = \log_b(x) + log_b(y)`$|$`\log_2(2 \times 3) = \log_2(2) + \log_2(3)`$| +|Quotient Law|$`\log_b(\dfrac{x}{y}) = \log_b(x) - \log_b(y)`$|$`log_2(\dfrac{2}{3}) = \log_2(2) - \log_2(3)`$ + +## Solving Logarithms + +1. Using a common base and equating the the 2 exponents to one another ($`2^x = 4^{x-5} \implies 2^x = 2^{2x-10} \implies x = 2x-10`$) +2. Using a log rule to simplify and bring the exponenets to the "living room/main floor". + +## Application of Exponential Growth + +The formula for **Exponential Growth** is given as: + +```math +\LARGE + +N = N_0(R)^{\frac{t}{d}} + +``` + +$`N = `$ Final amount. + +$`N_0 = `$ Starting amount. + +$`R =`$ Growth factor. +- $`R = 1 + r`$ +- **half-life:** $`R = \dfrac{1}{2}`$ +- **doubling time:** $`R = 2`$ + +**Growth Rate** +- $`r > 0`$ Exponential Growth +- $`-1 \lt r \lt 0`$ Exponential Decay +- r is ually given as a $`\%`$ + +$`t = `$ Total amount. (time for $`N_0`$ to get to $`N`$) + +$`d = `$ Growth Rate time. (Time for 1 Growth Rate to occur).