highschool/Grade 10/Math/MCR3U7/Unit 1: Exponential and Logarithmic Functions.md

Unit 1: Exponential and Logarithmic Functions

Review

Function: 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 $`(\dfrac{1}{k}x+d, ay+c)`$.

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 function 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:

\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 usually given as a percentage ($`\%`$)

$`t = `$: Total time for $`N_0`$ to get to $`N`$

$`d = `$ Time for 1 growth rate to occur