The population started at 10 elves in 2000, and grows by 5 each year to reach 110 in 2020. Note that the domain of this problem specifically prevents me from letting $x$ be negative (the "since 2000" comment), so we don't have to worry about interpreting something like $x=-20$ which could mean there are -10 elves in 1980.

The code below graphs the function.

p <- function(x,a=2,b=30){a*x+b}

x <- seq(-4,200,0.1)
plot(x,p(x,5,10),type='l',ylim=c(0,500))
abline(h=0,col='gray')
abline(v=0,col='gray')

p(20,5,10)

1. Because $2^3=8$, we know $x=3$.
2. Because $2^{-4}=\frac{1}{16}$, we know $x=-4$.
3. This one isn't guessable. We know $2^4=16$ and $2^5=32$, so we know $4<x<5$. How do we get the solution? This is what logarithms are for. We'll discuss a guessing approach in class, and then define logarithms as the way to tell the computer to preform this guessing approach. We can rewrite the exponential form $2^x = 17$ in the log form $x = \log_2(17)$, which is the solution.
4. Anything nonzero, raised to the zero, is 1, so the solution is $x=0$.
5. Rewriting in log form gives us $\log_e(2)=x$. We use $\ln$ as shorthand for $\log_e$.

In general, we rewrite $a^\text{input} = \text{output}$ in log form as $\log_a\text{output} = \text{input}$. Exponential and logarithmic functions are inverses of each other. They undo what the other function did, which is why the inputs and outputs swap spots.

Most computer languages use $\exp(x)$ for $e^x$ and $\log(x)$ for $\log_e(x)$. When using $\exp$ and $\log$, we can remember the rewriting rule as $$y = \exp(x) \quad \text{means} \quad x = \log(y).$$ Notice that when the inputs and outputs swap, the function changes from $\exp$ to $\log$ or vice versa.

An exponential function is of the form $b^x$ where $b>0$ and $b \neq 1$.

Note: An exponential function is not an algebraic function.

• Domain is all real numbers
• Range is positive real numbers
• When $b > 1$ the function is increasing on its entire domain.
  • As $x$ decreases (approaches negative infinity) the output gets close to zero.
  • As $x$ increases the output increases without bound.
  • These functions are concave up. (We will learn the precise meaning of concavity later this semester.)
• When $0< b < 1$ the function is decreasing on its entire domain.
  • As $x$ decreases (approaches negative infinity) the output grows without bound.
  • As $x$ increases the output gets close to zero.
  • These functions are concave up. (We will learn the precise meaning of concavity later this semester.)

If $a > 0$, $b >0$, and $m$ and $n$ are any real number, then

• $a^m \cdot a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $\frac{1}{a^n} = a^{-n}$
• $\sqrt[n]{a} = a^{1/n}$
• $\sqrt[n]{a^m} = a^{m/n}$
• $a^m \cdot b^m = (a \cdot b)^m$
• $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$

A logarithm base $b$ of a positive number satisfies the following definition.

For $x>0$, $b>0$, and $b \neq 1$, $y = \log_b(x)$ is equivalent to $b^y = x$

• We read $\log_b(x)$ as, "the log base $b$ of $x$" or "the logarithm with base $b$ of $x$".
• The logarithm is the exponent that $b$ must be raised to get $x$.
• When we evaluate $\log_b(x)$ we answer the question, "To what exponent must $b$ be raised to in order to get $x$?"

A logarithmic function is of the form $\log_b(x)$ where $b>0$ and $b \neq 1$.

Note: A logarithmic function is not an algebraic function.

• Domain is positive real numbers
• Range is all real numbers
• When $b > 1$ the function is increasing on its entire domain.
  • As $x$ gets close to zero from the right the output approaches negative infinity.
  • As $x$ increases the output increases without bound.
  • These functions are concave down.
• When $0< b < 1$ the function is decreasing on its entire domain.
  • As $x$ gets close to zero from the right the output grows without bound (approaches infinity).
  • As $x$ increases the output approaches negative infinity.
  • These functions are concave up.

If $a>0$, $b > 0$, $c > 0$, $b \neq 1$, and $n$ is any real number, then

• $\log_b b^x = x$ and $b^{\log_b x} = x$ (inverse property)
  • Special Cases
    • $\log_b b = \log_b b^1 = 1$
    • $\log_b 1 = \log_b b^0 = 0$
• $\log_b(ac) = \log_b(a) + \log_b(c)$ (product property)
• $\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)$ (quotient property)
• $\log_b(a^n) = n\log_b(a)$ (power property)