7.5 Chapter 7 Study Guide

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Definitions

exponential function: a function of the form $f(x) = a^x$, where $a >0$ and $a \neq 1$.
base (of an exponential function): the number $a$ being raised to the power $x$ in an exponential function.
end behavior: how a function behaves "eventually," as $x \rightarrow \pm \infty$.
logarithmic function: a function of the form $ f(x) = \log_a(x)$, where $a >0$ and $a \neq 1$, whose values are defined by the relationship
\[ \log_a x = y \quad \iff \quad a^y = x. \notag \]
natural log: the logarithmic function with base $e$.
expanding: the process of taking a single logarithm and writing it as the sum/difference of multiple simpler logarithms, including bringing powers down as coefficients if possible.
condensing: the process of taking a sum/difference of multiple logarithms and combining them into a single log using the Log Laws.
compound interest: interest that is calculated and paid on an investment every certain amount of time. Starting with initial investment amount $P$, with nominal annual rate $r$ written as a decimal, but compounded $n$ times a year, after $t$ years, you will have
\[ A(t) = P \left( 1 + \frac{r}{n} \right)^{nt} \notag \]
population doubling: population growth modeled by an exponential function with base 2. Starting with population initial size $n_0$, doubling every $a$ seconds/minutes/days/etc., the population at time $t$ will be
\[ n(t) = n_0 2^{t/a} \notag \]
relative growth rate: the rate $r$ of growth expressed as a proportion of the population at any time, which appears when modeling growth using base $e$ instead of $2$ like above. In this case, the function takes the form
\[ n(t) = n_0 e^{rt} \notag \]
logistic growth: a population exhibits logistic growth if it can be modeled using a logistic function:
\[ n(t) = \frac{M}{1+\frac{M-n_0}{n_0} e^{-rt}}\notag \]
carrying capacity: the maximum population an environment can support due to limiting factors. This is the $M$ in the logistic function above.
half-life: the amount of time it takes for a substance to decay to half its original amount.
radioactive decay: the decay of substances as they emit radiation over time. Starting with a mass $m_0$ and half-life $h$, the mass remaining at time $t$ is modeled with the function
\[ m(t) = m_0 e^{-rt} \notag \]
where $r$ is the relative decay rate, which can be computed from $h$ using $r = \frac{\ln 2}{h} $.
Newton's Law of Cooling: the fact that an object with initial temperature $T_0$ placed in an environment with surrounding temperature $T_s$ will approach the environment's temperature, with temp at time $t$ given by
\[ T(t) = T_s + (T_0 - T_s)e^{-rt} \notag \]
where $r$ is the coefficient of heat transfer.

Exponential Functions

The exponential function with base $a$ is defined by $ f(x) = a^x$, for all real numbers $ x$, where $ a > 0$ and $a \neq 1$.

Fun facts about exponential functions:

The domain of an exponential function is all real numbers.
If $a > 1$, the function is increasing (graph going uphill from left to right). If $ 0 < a< 1$, the function is decreasing (graph going downhill from left to right). See graphs below.
End behavior for $a > 1$: As $x \rightarrow -\infty$, the graph hugs a horizontal asymptote at the $x$-axis. As $x \rightarrow \infty$, the graph shoots up rapidly, and the bigger the base, the faster the growth.
End behavior for $ 0<a<1$: As $x \rightarrow -\infty$, the graph is blowing up to the left. As $x \rightarrow \infty$, the graph hugs the horizontal asymptote at the $x$-axis forever.
All exponential functions pass through $ (0,1)$ as well as $ (1,a)$, for whatever their base $a$ is.
The function values are always positive, aka above the $x$-axis, aka their range is $ (0, \infty) $.
For $ f(x) = a^x$ and $ g(x) = \left( \frac{1}{a}\right)^x $, note that $ g(x) = \left( \frac{1}{a}\right)^x = \frac{1}{a^x} = a^{-x} = f(-x) $.

$exp1.png$ $exp2.png$

Sketching the Graph of an Exponential Function

Find the $y$-intercept by plugging in $x = 0$ and drop that point in.
Compute a couple more easy-to-evaluate points, like plugging in $x = \pm 1, \pm 2$.
Check if there's a "$+k$" of some sort on the end of the whole function. That'll be a vertical shift. You could draw in your horizontal asymptote.
Check if there's a negative somewhere causing a reflection.
Use all that knowledge to connect the dots in the general shape you expect for an exponential, so...a big swoop.

Example: this is $f(x) = 10^{x+1} $. No vertical shift, so H.A. stays at $x$-axis.

$c.png$

Logarithmic Functions

For $a > 0$ and $ a \neq 1$, the logarithmic function with base $a$, which we denote $ \log_a$, is the inverse function of the exponential function with the same base $a$. Thus, it is defined by the relationship

\[ \log_a x = y \quad \iff \quad a^y = x. \notag \]

We say, "log base $a$ of $x$ is $y$." Basically, "$ \log_a (x) = ?$" is asking the question, "What do I need to raise the base $a$ to in order to get out this $x$?" The answer to that question is the output of the function $ \log_a$.

When in particular the base is the special number $e$, we write $ \log_e x $ as $ \ln x$, and call it the natural log of $x$.

Fun facts about logarithmic functions:

Their domain is only positive real numbers! You are not allowed to put 0 or a negative number into a log, because that would be the same as saying $ a^{y} = 0 $, and there's no possible $y$ that will satisfy that!
Since $a^0 = 1$ for any $a$, we have $ \log_a (1) = 0$ for any base $a$. This is something you should know without using a calculator.
Similarly, since $a^1 = a $, we have $ \log_a(a) = 1$ for any $a$. Aka, if you take the log of its own base, you will always get 1. This is the other evaluation you should know.
Inverse functions relationship: $ \log_a (a^x) = x,$ for all real numbers $ x $, and $ a^{\log_a x} = x,$ for all $ x> 0 $

$logs graphs.png$

Sketching the Graph of a Logarithmic Function

First identify the OG log function if you're looking at a transformation. We'll sketch that first.
1. Remember you have a vertical asymptote, the $y$-axis.
2. Plot the points $ (1,0) $ and $ (a,1)$, where $a$ is your base.
3. Draw a swoosh through the two points, hugging the $y$-axis forever going down, and slowly growing off as $x \rightarrow \infty$.
Identify any transformations that happened to make your desired function. Adjust the signal points you plotted accordingly, draw the new vertical asymptote if any horizontal shifts happened, and connect the dots nicely.

Example: this is $f(x) = \log_2 (x-1)+2 $. The dotted blue graph is the OG, $ \log_2 x$

$sk3.png$

Logarithm Laws

Log Law	English Translation	Caveman Translation
$ \log_a (AB) = \log_a A + \log_a B $	The logarithm of a product of expressions (numbers, variables, whatever as long as it is being multiplied) is the sum of the logarithms of the expressions.	Multiplication inside the log can be written as addition outside the log. Used backwards, addition outside the log translates to multiplication inside the log.
$ \log_a \left( \frac{A}{B} \right) = \log_a A - \log_a B $	The logarithm of a quotient of expressions is the difference of the logarithms of the expressions.	Division inside the log can be written as subtraction outside the log. Used backwards, subtraction outside the log translates to division inside the log. The denominator must be the input whose log term is subtracted!
$ \log_a ( A^B) = B \log_a A $	The logarithm of an expression raised to a power is the power times the logarithm of the expression.	If the whole input is raised to a power, you can bring powers down in front as coefficients.
$ \log_b x = \dfrac{ \log_a x}{\log_a b} $	Change of Base Formula: to express $\log_b x$ in terms of $ \log_a x$, divide (\log_a x\) by the constant $ \log_a b$.	Maybe you want to replace $ \log_b x$ with an equivalent expression that uses base $a$ instead. You can switch from $\log_b$ to $ \log_a$, but you have to divide by the number $ \log_a b$.

Note: The bases must match in order to use the first two laws backwards! Using them forwards is called expanding, and using them backwards is called condensing.

How to Solve Exponential Equations

Get everything boiled down to a single base-to-a-power term on one side of the equals.
Take the (appropriate) log of both sides and use the bring-powers-down Log Law.
Solve for the variable as usual. Check your answer by plugging back in. There may or may not be a valid solution!

How to Solve Logarithmic Equations

Condense completely until you get one single logarithm term on one side by itself.
Raise the base to each side of the equation (or translate to exponential form of the logarithmic equation).
Solve for the variable.

The best way to study application problems is to review the examples and exercises carefully, since there's such a variety!

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