Section Exponential and Logarithmic Functions
¶The goal of this appendix is to review exponential and logarithmic functions. Even though you should definitely have seen these functions before you made it to Calculus, it is my experience that many students benefit from a review of the basics. Use this as a review and ask if you have any troubles.
Problem 14.17.
Use a limit table to convince yourself that the \(\lim_{n \rightarrow \infty} (1 + \frac{1}{n})^n\) exists by substituting in \(n=10, n=100,\) and \(n=1000.\)
Definition 14.18.
We define the real number \(e\) by \(e = \lim_{n \rightarrow \infty} (1 + \frac{1}{n})^n.\)
Definition 14.19.
Any function of the form \(f(x) = b^x\) where \(b\) is any positive real number other than \(1\) is called an exponential function.
Problem 14.20.
Graph \(r(x) = 2^x, s(x) = 3^x, u(t) = 2^{-t},\) and \(v(t) = 3^{-t}\) all on the same pair of coordinate axes by either plotting points (preferable) or by using your favorite technological weapon. Pay special attention to the domain and range, x- and y-intercepts, and any vertical or horizontal asymptotes.
Definition 14.21.
The function \(f(x) = e^x\) is called the natural exponential function.
Problem 14.22.
Graph each of the following variants of the exponential function, listing domain, range, intercepts, and asymptotes.
\(f(x) = e^x\)
\(g(x) = e^{-x}\)
\(h(x) = e^{x-2}\)
\(i(x) = e^x-3\)
Definition 14.23.
We define the inverse of \(f(x) = e^x\) by \(g(x) = \ln(x)\) and refer to this as the natural logarithm function.
Definition 14.24.
We define the inverse of \(f(x) = b^x\) by \(g(x) = \log_b(x)\) and refer to this as the logarithm to the base b.
By definition of inverses, we have these Inverse Properties.
\(\ln(e^x) = x\) for all \(x \in \re\) and
\(e^{\ln(x)}=x\) for all \(x > 0.\)
The following theorem relates the natural logarithm function to all other base logarithm functions which means that if you really understand the natural logarithm function then you can always convert the other logarithms to the natural logarithm function.
Theorem 14.25.
Change of Base Theorem. For any \(b \in {\nat}\text{,}\) \(\dsp \log_b(x) = \frac{\ln(x)}{\ln(b)}.\)
Theorem 14.26.
Algebra of Exponential Theorem.
\(e^xe^y = e^{x+y}\)
\(e^x/e^y = e^{x-y}\)
\((e^x)^y = e^{xy}\)
Each of these laws leads to a corresponding statement about logarithms.
Theorem 14.27.
Algebra of Natural Log Theorem.
\(\ln(xy) = \ln(x) + \ln(y)\)
\(\ln(x/y) = \ln(x) - \ln(y)\)
\(\ln(x^y) = y\ln(x)\)
Problem 14.28.
Prove the Logarithmic Laws using the Exponential Laws and the Inverse Properties.
Problem 14.29.
Evaluate each of the following without using a calculator.
If \(e^4 = 54.59815 . . .\text{,}\) then \(\ln(54.59815 . . .) =\) .
If \(\ln(24) = 3 .1780538 . . .\text{,}\) then \(e^{3 .1780538} =\) .
\(e^{\ln(4)} =\) .
\(\ln(e^{x}) =\) .
Problem 14.30.
Evaluate each of the following using a calculator.
\(\ln(3.41) =\) .
\(e^{4\ln(2)} =\) .
Problem 14.31.
Write each of the following as a single logarithm.
\(3\ln(2x) - 2\ln(x) + \ln(y) - \ln(z)\)
\(2\ln(x+y) + \ln(\frac{1}{x-y})\)
\(\ln(x) + 3\;\;\;\) First fill in the blank: \(\;\;\;3 = \ln(\underline{\hspace{4.545454545454546em}}).\)
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Which of the following is correct? Why?
\(\dsp \ln(x) - \ln(y) + \ln(z) = \ln(\frac{x}{yz})\) or
\(\dsp \ln(x) - \ln(y) + \ln(z) = \ln(\frac{xz}{y})\)
Problem 14.32.
Expand each of the following into a sum, difference, or multiple of the logarithms.
\(\ln(9x^2y)\)
\(\ln(4x^{-1}y^2)\)
\(\dsp \ln(\frac{xy^2}{z^3})\)
\(\dsp \ln\big( \frac{ (x+y)^2 }{ (x-y)^2 } \big)\)
Problem 14.33.
Solve each of the following for x; give both exact and approximate answers.
\(\ln(3x) = 2\)
\(\ln(6) + \ln(2x) = 3\)
\(\ln(x^2-x-5)=0\)
\(\ln(x+3) + \ln(x-2)=2\)
\(\ln ^2x-\ln x^5 + 4 = 0\) Notation: \(\ln ^2x\) means \(\ln(x) )^2\) and \(\ln x^5\) means \(\ln(x^5)\)
\(\ln ^2x+\ln x^2-3=0\)
Problem 14.34.
Solve for x; give both exact and approximate answers.
\(e^x = 19\)
\(4^x = \frac{2}{3}\)
\(4^{x+1} = e\)
\((\frac{1}{2})^x = 3\)
\(x^2 e^x - 9 e^x = 0\) Factor, factor, factor!
\(e^x - 8e^{-x} = 2\)
\(5^x = 4^{x+1}\)
\(3^{2x+1} = 2^{x-1}\)
Solutions to Exponential and Logarithmic Exercises
\begin{sol} These are the solutions to Problem 14.29.
\(4\)
\(24\)
\(4\)
\(x\)
\end{sol}
\begin{sol} These are the solutions to Problem 14.30.
\(1.2267123\)
\(16\)
\end{sol}
\begin{sol} These are the solutions to Problem 14.31.
\(\dsp \ln(\frac{8xy}{z})\)
\(\dsp \ln(\frac{(x+y)^2}{(x-y)})\)
\(\ln(xe^3)\)
Only the second statement is true.
\end{sol}
\begin{sol} These are the solutions to Problem 14.32.
\(\ln(9) + 2\ln(x) + \ln(y)\)
\(\ln(4) + 2\ln(y) - \ln(x)\)
\(\ln(x) + 2\ln(y) - 3\ln(z)\)
\(2\ln(x+y) - 2\ln(x-y)\)
\end{sol}
\begin{sol} These are the solutions to Problem 14.33.
\(e^2/3\)
\(e^3/12\)
\(-2,3\)
\(\dsp \frac{-1+\sqrt{25+4e^2}}{2}\)
\(e, e^4\)
\(\frac{1}{e^3},e\)
\end{sol}
\begin{sol} These are the solutions to Problem 14.34.
\(ln19\)
\(\dsp \frac{\ln(\frac{2}{3})}{\ln4}\)
\(\dsp \frac{1}{\ln4}-1\)
\(\dsp \frac{\ln3}{\ln\frac{1}{2}}\)
\(-3,3\)
\(ln4\)
\(\dsp \frac{\ln4}{\ln(\frac{5}{4})}\)
\(\dsp \frac{\ln\frac{1}{6}}{\ln\frac{9}{2}}\)
\end{sol}