Before we look at logarithmic functions, let’s quickly review exponents and logs. For a particular base, let’s say 5, taking a logarithm is the opposite operation for raising to a power. For example, if we raise base 5 to a power of \(\alert{2}\text{,}\) we get
We can say that a logarithm is actually an exponent. Asking for the log base 5 of 25 is asking "what power of 5, or what exponent on base 5 will give me 25?"
is a logarithmic function. In order to understand logarithmic functions better, we first investigate how they are related to more familiar functions, the exponential functions.
You know that raising to the \(n\)th power and taking \(n\)th roots are inverse operations. For example, if we first cube a number and then take the cube root of the result, we return to the original number.
\begin{align*}
x \amp = \alert{5}~~~~~~\rightarrow~~~~~~x^3=125~~~~~~\rightarrow~~~~~~\sqrt[3]{x^3}=\sqrt[3]{125}=\alert{5}\\
\amp \blert{\text{Cube the number}}~~~~~~~~\blert{\text{Take the cube root}}~~~~~~~~\blert{\text{Original number}}
\end{align*}
Now consider the functions that describe those operations: The graphs of \(f(x)=x^3\) and \(g(x)=\sqrt[3]{x}\) are related in an interesting way, as shown below.
If we place a mirror along the line \(y=x\text{,}\) each graph is the reflection of the other. Or imagine folding the grid along the line \(y=x\text{,}\) and one graph will match the other exactly. We say that the graphs are "symmetric about the line \(y=x\text{.}\)"
By interchanging the rows in the table for \(f(x)\text{,}\) we get the table for \(g(x)\text{.}\) This makes sense when we recall that each function undoes the effect of the other. In fact, we define the cube root function by
\begin{equation*}
y=\sqrt[3]{x}~~~~\text{if and only if}~~~~x=y^3
\end{equation*}
In other words, if we interchange the variables in the cubing function, \(y=x^3\text{,}\) to get \(x=y^3\text{,}\) we have an equivalent formula for the cube root function, \(y=\sqrt[3]{x}\text{.}\)
Let’s look at inverse operations in terms of functions. If we apply a function and then its inverse to an input, we return to that input. Use \(f(x)=x^3\) and \(g(x)=\sqrt[3]{x}\text{,}\) for example, and start by applying the innermost function.
Without the function notation we are simply saying that \(\sqrt[3]{x^3} = x\text{.}\) But this is an example of the more general rule that if \(f(x)\) and \(g(x)\) are inverse functions, then \(g(f(x))= x\) and \(f(g(x)) = x\text{.}\)
A similar rule relates the operations of raising a base \(b\) to a power and taking a base \(b\) logarithm. You may recall from Chapter 7 that
\begin{equation*}
y = \log_b {(x)}~~~~\text{if and only if}~~~~x=b^y
\end{equation*}
So the function \(g(x)=\log_b {(x)}\) is the inverse function for \(f(x)=b^x\text{.}\) Each function undoes the effect of the other. For example, let \(f(x)=2^x\) and \(g(x)=\log_2 {(x)}\text{.}\) Start with \(x=3\text{,}\) apply \(f\text{,}\) and then apply \(g\) to the result.
\begin{equation*}
x = \alert{3}~~~~~~\rightarrow~~~~~~f(3)=2^3=8~~~~~~\rightarrow~~~~~~g(8)=\log_2{(8)} = \alert{3}
\end{equation*}
(Remember the order of operations: do what’s inside of parentheses first, to get \(\log_2{(8)}\text{.}\)) Because the log and the exponential are inverse functions, this identity holds for any value of \(x\) and any base \(b \gt 0\text{.}\) That is
Because \(\log {(x)}\) and \(10^x\) are inverse functions, \(\log {(10^{3.4})} = 3.4\text{.}\) You can verify the result on your calculator by first computing \(10^{3.4}\) and then taking the log of the result.
Again, because \(\log {(x)}\) and \(10^x\) are inverse functions, \(10^{\log{(0.6)}} = 0.6\text{.}\) You can verify the result on your calculator by first computing \(\log {(0.6)}\) and then computing \(10\) to the result.
We plot the points for each function, connecting them with smooth curves, as shown below. You can see that the two graphs are symmetric about the line \(y=x\text{.}\)
Did you notice that the graphs of the log functions do not have any points with negative \(x\)-coordinates? This is because an exponential function has no negative (or zero) output values, so a log function has no negative or zero input values.
For example, suppose that \(~\log_5{(-25)} = x~\text{.}\) Then \(~5^x = -25~\text{.}\) But this is impossible, because \(5^x\) cannot be negative. Thus, the log of a negative number (or zero) is undefined. In other words,
You can also see that while an exponential growth function increases very rapidly for positive input values, its inverse, the logarithmic function, grows extremely slowly.
\begin{equation*}
T = \dfrac{\log {(2)} \cdot t_i}{3 \log{(D_f /D_0)}}
\end{equation*}
is used by X-ray technicians to calculate the doubling time of a malignant tumor. \(D_0\) is the diameter of the tumor when first detected, \(D_f\) is its diameter at the next reading, and \(t_i\) is the time interval between readings, in days.
Life expectancy at birth is the average number of years a newborn child is expected to live. In 1900, the average life expectancy at birth in the U.S. was 47.3 years, and in 1910 it had risen to 50.0 years. During rest of the twentieth century, life expectancy was modeled by the formula
We can make a table of values for \(L(x)\) and plot points to obtain the graph below, which also shows the actual data points for life expectancy for the decades from 1910 to 2000.
In the previous Example, we see that although life expectancy has been increasing over time, it has been slowing down or leveling off. In fact, life expectancy in the US actually declined slightly from 78.94 in 2013 to 78.81 in 2018. (What factors may have contributed to this decline?) By 2024 it had rebounded to 79.25. It remains to be seen how well the model predicts life expectancy in the 21st century.
The CDC (Centers for Disease Control and Prevention) provides Growth Charts for the average height and weight of children from age 2 to 20. The average height of girl children is given in centimeters by
If there is only one log involved, we can use the conversion equations to write the equation in exponential form. (Recall the conversion equations from Section 7.3.1.)
But \(x = -3\) is not a solution of the original equation, because neither \(\log_{10}{(x + 1)}\) nor \(\log_{10}{(x - 2)}\) is defined for \(x = -3\text{.}\) (Remember that we cannot take a logarithm of a negative number or zero.) We say that the apparent solution \(x=-3\) is extraneous, and the only solution of the original equation is \(4\text{.}\)
Extraneous solutions can arise whenever we solve a logarithmic equation, especially if there is more than one apparent solution. Therefore, we should always check that a possible solution does not cause one of the logarithms to be undefined.
In a psychology experiment, volunteers were asked to memorize a list of nonsense words, and 24 hours later they were tested to see how many of the words they recalled. On average, the subjects had forgotten 20% of the words. The researchers found that the more lists their volunteers memorized, the larger the fraction of words they were unable to recall. (Source: Underwood, Scientific American, vol. 210, no. 3)
The water velocity at any point in a stream or river is related to the logarithm of the depth at that point. For the Hoback River near Bondurant, Wyoming,
\begin{equation*}
v = 2.63 + 1.03 \log {(d)}
\end{equation*}
where \(v\) is the velocity of the water, in feet per second, and \(d\) is the vertical distance from the stream bed, in feet, at that point. For Pole Creek near Pinedale, Wyoming,
\begin{equation*}
v = 1.96 + 0.65 \log {(d)}
\end{equation*}
Both streams are \(1.2\) feet deep at the locations mentioned. (Source: Leopold, Luna, Wolman, and Gordon, 1992)
The average velocity of the entire stream can be closely approximated as follows: Measure the velocity at 20% of the total depth of the stream from the surface and at 80% of the total depth, then average these two values. Find the average velocity for the Hoback River and for Pole Creek.
