Section 1.3 Logarithmic Functions
References References
- Calculus, Early Transcendentals by Stewart, Section 1.5 (the second part).
Does an exponential function \(y=f(x)=a^{x}\) have an inverse?
For \(a>1\text{,}\) the value of \(a^x\) increases as \(x\) increases: the graph is increasing, which is enough to pass the HLT and ensure existence of an inverse.
For \(0 \lt a \lt 1\text{,}\) the graph is decreasing and so again passes the HLT, giving an inverse. (For \(a=1\text{,}\) there is no inverse, and the function is boring: \(f(x) = 1\text{.}\))
This inverse should be familiar: the number \(x\) for which \(a^x=y\) is called the logarithm of \(y\) base \(a\), written \(\log_{a}y\text{,}\) so the inverse of the exponential function \(f(x)=a^{x}\) is the logarithmic function base \(a\), \(f^{-1}(x)=\log_{a}x\text{.}\)
An exponential function for \(a \neq 1\) is defined for all real numbers (its domain) and its values (range) are all positive numbers. Thus the logarithmic functions \(\log_{a}\) have domain all the positive numbers, range all the reals: only positive numbers have logarithms.
Note that this simple domain and range for logarithms depends on exponential functions being defined for all real arguments, not just all rational arguments.
Rules for logarithms.
Logarithms satisfy the following rules, all following from the rules for exponentials in Section 1.5; For any positive number \(a\) except 1, and any positive numbers \(x\) and \(y\text{,}\)- \(\displaystyle \log_a(x \cdot y) = \log_a x + \log_a y\)
- \(\displaystyle \log_a(x/y) = \log_a x - \log_a y\)
- \(\log_a(x^p)=p \log_a x\) for any real power \(p\text{.}\)
Natural logarithms.
Since \(e^{x}\) is the most commonly used exponential function, its inverse \(\log_{e}\) is the most important logarithmic function: it is called the natural logarithm, and has the special name \(\ln\) (from the initials of “logarithm” and “natural”):
\begin{equation*}
\ln x = \log_e x.
\end{equation*}
Our first use of the natural logarithm is to put any exponential function \(a^{x}\) in terms of \(e^{x}\text{.}\) Using the properties of exponentials and the fact that \(e^{\ln a}=a\text{,}\)
\begin{equation*}
a^x = (e^{\ln a})^x = e^{(\ln a)x}.
\end{equation*}
It can also be shown that
\begin{equation*}
\log_{a} x = \frac{\ln x}{\ln a},
\end{equation*}
so we can also put all logarithmic functions in terms of natural logarithms.
Thus we mostly need just one exponential function, \(e^{x}\text{,}\) and just one logarithmic one: its inverse, the natural logarithm.
For now we omit the topic Inverse trigonometric functions; instead we will review those when we encounter them in Section 3.7.
Exercises Exercises
Study Calculus Volume 1, Section 1.5 2 , Exercises 247, 251, 255, 261, 265, 271, 273, 277, 283, 285, 287, and 305(b).
openstax.org/books/calculus-volume-1/pages/1-5-exponential-and-logarithmic-functionsopenstax.org/books/calculus-volume-1/pages/1-5-exponential-and-logarithmic-functions
