3.5 KiB
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
- \(`\log_b(b^x) = x`\)
- \(`b^{\log_b(x)} = x`\)
- \(`\log_b(1) = 0`\)
- \(`\log_b(b) = 1`\)
| 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
- Using a common base and equating the the 2 exponents to one another (\(`2^x = 4^{x-5} \implies 2^x = 2^{2x-10} \implies x = 2x-10`\))
- 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