Section 4 Logarithmic Functions and Their Modeling
Given two functions \(y=f(x)\) and \(z=g(y)\text{,}\) we can construct a new function \(z = (g \circ f)(x) = g(f(x))\text{,}\) which is called the composition of \(g\) and \(f\text{.}\)
Task 4.2.
Let \(f(t)=t^2+1\) and \(g(s) = 2s\text{.}\)- Find the function equation of \((g \circ f)(x)\text{.}\)
- Find the function equation of \((f \circ g)(x)\text{.}\)
- Is \((g \circ f)(x)\) the same function as \((f \circ g)(x)\text{?}\) What does this tell you about the order of composition of two functions?
Task 4.3.
Let \(f(m)=2m+1\) and \(g(p) = \frac{1}{2}p-\frac{1}{2}\text{.}\)- Find the function equation of \((g \circ f)(x)\text{.}\)
- Find the function equation of \((f \circ g)(x)\text{.}\)
- Is \((g \circ f)(x)\) the same function as \((f \circ g)(x)\text{?}\)
Given a function \(f\) defined on the domain \(D\) with the range \(E\) and another function \(g\) defined the domain \(E\) with the range \(D\text{.}\) We say that \(g\) is the inverse function of \(f\) if \(g(f(x))=x\) for all \(x\) in \(D\) and \(f(g(x))=x\) for all \(x\) in \(E\text{.}\) Often the inverse function \(g\) is denoted by \(f^{-1}\text{.}\)
Task 4.4.
Let \(z(t)=\sqrt{t-3}\) be a function.- What are the domain and the range of this function?
- Find a function equation for the inverse function \(z^{-1}(t)\text{.}\)
- What are the domain and the range of \(z^{-1}\text{?}\)
Task 4.5.
Let \(h(x)=3x-7\) be a function.- What are the domain and the range of this function?
- Find a function equation for the inverse function \(h^{-1}(t)\text{.}\)
- What are the domain and the range of \(h^{-1}\text{?}\)
The inverse function of an exponential function \(f(x)=b^x\) is called the logarithmic function \(g(x)=\log_b(x)\text{.}\) According to the definition of inverse functions, we know that \(f(g(x)) = x\) and \(g(f(x))=x\text{.}\) Therefore we have
\begin{equation*}
b^{\log_b(x)}=x, \qquad \log_b(b^x)=x.
\end{equation*}
This tells us that the number \(\log_b(x)\) is the exponent of \(b\) that gives \(x\text{,}\) that is, if \(y=\log_b(x)\text{,}\) then \(b^y=x\text{.}\) When the base number \(b=e\text{,}\) we write \(\log_b(x)\) as \(\ln(x)\) and call it the natural logarithmic function. We have
\begin{equation*}
\ln(e^x) = x, \qquad e^{\ln(x)}=x.
\end{equation*}
Task 4.6.
- What is the domain and the range of the function \(y=e^x\text{?}\)
- What is the domain and the range of the function \(y=\ln(x)\text{?}\)
Task 4.7.
Use the definition of logarithms to simplify each expression.- \(\ln(\frac{1}{e})\text{.}\)
- \(e^{\ln(5)}\text{.}\)
- \(\frac{\ln(1)}{3}\text{.}\)
- \(2\ln(e)\text{.}\)
Task 4.8.
Solve the following equations- \(9 = 3^{x+1}\text{.}\)
- \((\frac{1}{2})^x=8\text{.}\)
Task 4.9.
Use your calculator and pick values for \(A, B\) and \(r\) to test whether each of the following is true. If the statement is false, cross it out so that you do not attempt to apply it.- For every positive numbers \(A, B\text{,}\) \(\log_b(AB) = \log_b(A) + \log_b(B)\text{.}\)
- For every positive numbers \(A, B\text{,}\) \(\log_b(A+B) = \log_b(A) + \log_b(B)\text{.}\)
- For every positive number \(A\) and every number \(r\text{,}\) \(\log_b(A^r) = r\log_b(A)\text{.}\)
- For every positive numbers \(A, B\text{,}\) \(\log_b(\frac{A}{B}) = \log_b(A) - \log_b(B)\text{.}\)
- For every positive numbers \(A, B\text{,}\) \(\frac{\log_b(A)}{\log_b(B)} = \log_b(A) - \log_b(B)\text{.}\)
- For every positive numbers \(A, B\text{,}\) \(\log_b(A - B) = \log_b(A) - \log_b(B)\text{.}\)
Note that \(\ln(x) = \log_e(x)\) so all the properties that hold for \(\log_b(x)\) for any base \(b\) also hold for \(\ln(x)\text{.}\)
Task 4.10.
Write each logarithm in terms of \(\ln(2)\) and \(\ln(3)\text{.}\)- \(\ln(6)\text{.}\)
- \(\ln(\frac{2}{27})\text{.}\)
Task 4.11.
Expand each logarithmic expression.- \(\log_4 (5x^3y)\text{.}\)
- \(\ln(\frac{\sqrt{3x-5}}{7})\text{.}\)
Task 4.12.
Condense each logarithmic expression.- \(\frac{1}{2}\log_3(x)+3\log_3(x+1)\text{.}\)
- \(2\ln(x+2)-\ln(x)\text{.}\)
Task 4.13.
Solve the following two equations that involve logarithmic functions.- \(5+2\ln(x)=4\text{.}\)
- \(2\log_5(3x)=4\text{.}\)
On the Richter scale, the magnitude \(R\) of an earthquake of intensity \(I\) is given by
\begin{equation*}
R = \log_{10} \frac{I}{I_0}.
\end{equation*}
Here \(I_0=1\) is the minimum intensity that can be detected and is used for comparison. Intensity is a measure of the wave energy of an earthquake.
Task 4.14.
Find the intensity of each earthquake:- North Bay, CA in August 24, 2014, \(R=6.0\) (Fatality: 1).
- China in May 12, 2008, \(R=8.0\) (Fatalities: 87587).
Common calculators can only calculate logarithm with base \(10\) or base \(e\text{.}\) What if we want to calculate logarithms that have bases different from \(10\) and \(e\text{?}\) We can use the following base changing formula:
\begin{equation*}
\log_b(x) = \frac{\log_a(x)}{\log_a(b)}.
\end{equation*}
Task 4.15.
- Calculate the value of \(\log_4(25)\) using the base changing formula with the new base \(10\text{.}\)
- Calculate the value of \(\log_4(25)\) using the base changing formula with the new base \(e\text{.}\)
- Compare your answers of the previous questions.
Task 4.16.
In order to see why the base changing formula is true, we will check that
\begin{equation*}
\log_a(b) \times \log_b(x) = \log_a(x).
\end{equation*}
- First, simplify the exponential \(a^{\log_a(x)}\text{.}\)
- Second, simplify the exponential \(a^{\log_a(b) \times \log_b(x)}\) by first simplifying \(a^{\log_a(b)}\text{.}\)
- What can you say about \(a^{\log_a(x)}\) and \(a^{\log_a(b) \times \log_b(x)}\text{?}\) What does that tell you about \(\log_a(x)\) and \(\log_a(b) \times \log_b(x)\text{.}\)
We have learn several categories of functions such as linear functions, quadratic functions, exponential functions, logarithm functions. When given a set of data, how do we decide which type of functions we choose to model the set of the data?
Task 4.17.
Go to the link https://www.desmos.com/calculator/lwopk4scwd 1 . There are three sets of the data. Determine which type of functions should be used to model each set of the data and the find the exact function equations.www.desmos.com/calculator/lwopk4scwd