day1.2-slides

EDS 212: Day 1, Lecture 2

Exponential functions, logarithms, graphs, average slope

August 5^th, 2024

Math brain warm up

Algebra blitz
Exponentials and logarithms
Common units and unit conversions
Functions
Understanding graphs
Interpreting equations

The Natural Exponential

In previous examples, we evaluated exponentials with different bases that were variables (e.g. $x^5$) and rational numbers (e.g. $2^4$).

Here, we’ll learn about the natural exponential, $e$, which appears frequently in environmental science and modeling.

Where does $e$ come from?

The value is from continuous compounding over infinite intervals:

\[\sum^{\infty}_{k=0} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...\]

The e is from Leonard Euler, Swiss mathematician who proved the value was irrational

$e$ is a number, not a variable

$e = 2.71828182845904523536...$

It is an irrational number, yes - meaning it can’t be expressed by a simple ratio of integers - but a number nonetheless. With infinite decimal places.

It is always the same value.

Why is $e$ so common?

One reason: Exponential trends show up a LOT in environmental science (the proportional change is the same over each time span)
Math reason: Turns out it’s a very useful value for calculus

    x            y  previous_y percent_change_y
1   1     2.718282          NA               NA
2   2     7.389056    2.718282         63.21206
3   3    20.085537    7.389056         63.21206
4   4    54.598150   20.085537         63.21206
5   5   148.413159   54.598150         63.21206
6   6   403.428793  148.413159         63.21206
7   7  1096.633158  403.428793         63.21206
8   8  2980.957987 1096.633158         63.21206
9   9  8103.083928 2980.957987         63.21206
10 10 22026.465795 8103.083928         63.21206

A real-world example:

But populations can’t grow exponentially forever . . .

Gause, G. F. 1934. The Struggle for Existence. Baltimore: Williams and Wilkins.

Logistic growth

\[N_t=\frac{K}{1+[\frac{K-N_0}{N_0}]e^{-rt}}\]

Where $N_t$ is the population size at time $t$, $K$ is the carrying capacity, $N_0$ is the initial population size, and $r$ is a growth rate.

We should always think about why an equations has the shape it has - both conceptually and mathematically.

Logistic growth

\[N_t=\frac{K}{1+[\frac{K-N_0}{N_0}]e^{-rt}}\]

Why might we expect logistic growth for many populations?
What variables besides time would influence the actual population?

Logarithms

Logarithms ask a question: $\log_a(b)$ asks “to what power do I have to raise $a$ to get a value of $b$?

For example:

$\log_2(8)$ asks “to what power do I have to raise 2, to get a value of 8?”
$\log_x(x^{5.9})$ asks “to what power do I have to raise $x$ to get a value of $x^{5.9}$?”
$\log_{banana}(banana^{1382.95})$ asks “to what power do I have to raise $banana$ to get a value of $banana^{1382.95}$?”

The natural log = “log base $e$” = $\log_e()$ = $ln()$

So based on what we learned in the previous slide, what is:

$\log_e(e^{10.4})$ = ?
$\log_e(e^{2x+8.3})$ = ?
$\log_e(e^{ax+2b^2-4.095})$ = ?

Some log rules

$\log_x(AB)=\log_x(A)+\log_x(B)$
$\log_x(\frac{A}{B}) = \log_x(A)-\log_x(B)$
$\log_x(A^B)=B\log_x(A)$
$x^{\log_x(A)}=A$
$\log_x(1)=0$
$\log_x(x)= 1$

Critical thinking questions

Can the value within a $ln()$ expression ever be 0, or negative? Why?

Can the solution to a natural log expression ever be negative? How?

Working with $ln()$ and $e$ in equations

We can think of these as inverses of each other:

$e^{\ln(x)} = x$
$\ln(e^x)=x$

…and use that as a tool for escaping variables from exponents & logs (remembering we can do whatever we want to an equation, as long as we do the same exact thing to both sides)

Examples:

Find $x$ given $\ln(x) = 5$

Exponentiate both sides: $e^{\ln(x)} = e^5$ ; simplify left-hand side to get: $x = e^5$

Find $y$ given $e^{3y}=95$

Take natural log of both sides: $\ln(e^{3y})=ln(95)$ ; simplifies to: $3y = \ln(95)$, so $y = \frac{\ln(95)}{3}$

Math brain warm up

Algebra blitz
Exponentials and logarithms
Common units and unit conversions
Functions
Understanding graphs
Interpreting equations

Graphs: visualizing & thinking about data

Graphs are a way for us to more easily process trends or patterns that may be more challenging to understand in a table or list.

When you look at graphs, the first things you should ask:

What variables are plotted (e.g. x- & y-axis, including units)?
What values are plotted (e.g. raw values, transformed, means, etc.)?
What are the overall takeaways and am I understanding them responsibly?

Practice saying these things OUT LOUD as if presenting the graph to an audience

“This figure shows the [change/pattern/relationship] between [x-variable], shown on the x-axis in units of [units] and [y-variable], shown on the y-axis in units of [units]. Overall [overall statement of pattern / trend / findings].”

Possibly with additional context as useful for the audience to put those findings into perspective (e.g. “this reduction represents an 82% decline in rainbowfish stocks along the Narnia Coast since 1991”).

Slope (average)

Sometimes, it can be useful to find the average rate of change of a function. Between any two points on a function $(x_1,y_1)$ and $(x_2,y_2)$ the slope is found by:}

\[m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}\]

Get into the practice of saying the meaning out loud

As if you’re explaining it to someone unfamiliar with the data
Including units
Without overstating certainty

For example:

“Between 1972 and 2020 the price of hobbit homes increased by an average of $2,450 per year”

differs from

“Between 1972 and 2020 the price of hobbit homes increased by $2,450 per year.”

The average slope of a continuous function rarely tells the whole story