Concavity.
The graph of a function is concave up if it bends upward as we move left to right; it is concave down if it bends downward. A line is neither concave up nor concave down.
Section 0.2 Exponential and Logarithmic Functions
In this chapter we define exponential and logarithmic functions. For a more extensive treatment of exponential functions we refer the reader to PreCalculus at Nebraska: Exponential Functions and for a more extensive treatment of exponential functions we refer the reader to PreCalculus at Nebraska: Logarithmic Functions
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mathbooks.unl.edu/PreCalculus/Exponential-Functions.html7
mathbooks.unl.edu/PreCalculus/section-33.htmlSubsection 0.2.1 Exponential Functions
Before we discuss exponential functions, it is first useful to define the notion of concavity.
An exponential function will be either be increasing or decreasing, but will always be concave up! We define such a function as follows:
Exponential Function.
\begin{equation*}
P(t) = a \cdot b^t \text{, where } b \gt 0 \text{ and } b \ne 1 \text{, } a \ne
0.
\end{equation*}
The constant \(a \) is the \(y\)-value of the \(y\)-intercept of the function.
The equation
\begin{equation*}
P(t)=P_0 a^t
\end{equation*}
gives an exponential function with base \(a\text{.}\) Then
\begin{equation*}
\frac{P(t+1)}{P(t)}=\frac{P_0 a^{t+1}}{P_0 a^t}=a =\text{constant rate of
growth/decay}
\end{equation*}
- Growth: \(a\gt1\text{:}\) Doubling time: the time it takes to double the initial amount
- Decay \(0\lt a\lt1\text{:}\) Half-life: the time it takes to decay to half of the initial amount
Doubling Time and Half-life.
- The doubling time of an exponentially increasing quantity is the time required for the quantity to double.
- The half-life of an exponentially decaying quantity is the time required for the quantity to be reduced by a factor of one half.
We will (naturally) consider exponentials (and later, logarithms) in base
\begin{equation*}
e \approx 2.718281828459... \text{ (irrational number)}
\end{equation*}
For example, the general exponential function in the natural base
\begin{equation*}
P(t)=P_0 a^t=P_0 (e^k)^t=P_0e^{kt}, \quad a=e^k.
\end{equation*}
Thus, we will have
- exponential growth if \(a\gt 1\text{,}\) which gives \(k\gt 0\)
- exponential decay if \(0\lt a\lt 1\text{,}\) which gives \(k\lt 0\)
The number \(k\) is called the continuous rate of growth/decay. To find \(k\) we will need the logarithmic function.
Subsection 0.2.2 Logarithmic Functions
Recall inverse functions. If for each \(y\) in the range of \(f\) there exists exactly one value of \(x\) such that \(f(x)=y,\) then \(f\) has an inverse at \(y\) denoted by \(f^{-1}\) such that
\begin{equation*}
f(x)=y \quad \iff \quad f^{-1}(y)=x
\end{equation*}
Hence
\begin{equation*}
f(f^{-1}(y))=y \quad \text{and} \quad f^{-1}(f(x))=x.
\end{equation*}
The inverse of the exponential function \(f(x)=e^x\) is the natural logarithmic function \(f^{-1}(x)=
\ln(x)\) so we have
\begin{equation*}
\ln(x) =c \iff e^c=x
\end{equation*}
The Logarithm.
Let \(b\neq 1 \) be a positive number, then the function
\begin{equation*}
f(t)=\log_b(t)
\end{equation*}
is called a logarithm with base \(b \text{.}\)
Upon inputting a value \(t\text{,}\) the function \(\log_b(t)\) will tell you the power of \(b \) which will yield \(t \text{.}\)
Due to the relationship between logarithms and exponentials, we often say that the equations
\begin{equation*}
x=\log_b(y)\ \ \ \ \text{and}\ \ \ \ b^x=y
\end{equation*}
are equivalent.
Warning 0.29.
Two common bases of a logarithm are base \(10 \) and \(e\text{.}\) Since they are used so often, we have developed a short hand notation for a logarithm of base \(10\) and a logarithm of base \(e\text{.}\) This short hand is shown below:
\begin{equation*}
\log_{10}(y)=\log(y).
\end{equation*}
\begin{equation*}
\log_{e}(y)=\ln(y).
\end{equation*}
In other words, we simply drop the subscript when referring to base \(10 \) and we change to \(\ln\) when referring to base \(e\)
There are some important properties of logarithms that you should be familiar with.
Properties of Logarithms.
If \(x,y,b \gt 0\text{,}\) and \(b\neq 1\text{,}\) then
- \(\displaystyle \log_b(xy)=\log_b(x)+\log_b(y),\)
- \(\displaystyle \log_b(\frac{x}{y})=\log_b(x)-\log_b(y),\)
- \(\displaystyle \log_b(x^k)=k\cdot \log_b(x),\)
- \(\displaystyle \log_b(b^y)=y,\)
- \(\displaystyle b^{\log_b(x)}=x.\)
Subsection 0.2.3 Supplemental Videos
- Introduction
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unl.yuja.com/V/Video?v=7114207&node=34303316&a=13030261&autoplay=1 - Half Life
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unl.yuja.com/V/Video?v=7114205&node=34303257&a=52467459&autoplay=1 - Logarithm
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unl.yuja.com/V/Video?v=7114203&node=34303274&a=166161527&autoplay=1 - Examples
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unl.yuja.com/V/Video?v=7114209&node=34303222&a=102937779&autoplay=1
Exercises 0.2.4 Exercises
1. General Exponential Functions.
If you write the function \(P=8 e^{2 t}\) in the form \(P=P_{0}a^{t}\text{,}\) then
\(P_0 =\) , and
\(a =\) .
This function represents exponential
- growth
- decay
- neither growth nor decay
.
2. Finding Exponential Functions.
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3. Finding the Parameters for an Exponential Function.
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4. Half-Life.
Find the half-life (in hours) of a radioactive substance that is reduced by \(5\) percent in \(65\) hours.
Half life = (include units)
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/webwork2_files/helpFiles/Units.html5. Applied Half-Life.
In the year 2005, a picture supposedly painted by a famous artist some time after 1595 but before 1645 contains 96 percent of its carbon-14 (half-life 5730 years).
From this information, could this picture have been painted by this artist?
- Yes
- No
Approximately how old is the painting? years
6. Solving Exponential Equations.
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7. Using Inverse Functions With Exponentials and Logarithms.
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