Subsection The Natural Exponential Function
The natural exponential function is the function \(f(x) = e^x\text{.}\) Values for \(e^x\) can be obtained with a calculator using the \(\boxed{e^x}\) key ( 2nd
LN
on most calculators). For example, you can evaluate \(e^1\) by pressing
2nd
LN
\(1\)
to confirm the value of \(e\) given above.
Because \(e\) is a number between \(2\) and \(3\text{,}\) the graph of \(f(x) = e^x\) lies between the graphs of \(y = 2^x\) and \(y = 3^x\text{.}\) Compare the tables of values and the graphs of the three functions below. As with other exponential functions, the domain of the natural exponential function includes all real numbers, and its range is the set of positive numbers.
\(x\) 
\(y=2^x\) 
\(y=e^x\) 
\(y=3^x\) 
\(3\) 
\(0.125\) 
\(0.050\) 
\(0.037\) 
\(2\) 
\(0.250\) 
\(0.135\) 
\(0.111\) 
\(1\) 
\(0.500\) 
\(0.368\) 
\(0.333\) 
\(0\) 
\(1\) 
\(1\) 
\(1\) 
\(1\) 
\(2\) 
\(2.718\) 
\(3\) 
\(2\) 
\(4\) 
\(7.389\) 
\(9\) 
\(3\) 
\(8\) 
\(20.086\) 
\(27\) 
Example 5.40
Graph each function. How does each graph differ from the graph of \(y = e^x\text{?}\)
 \(g(x) = e^{x+2}\)
 \(h(x) = e^x + 2\)
Solution
If \(f(x) = e^x\text{,}\) then \(g(x) = f(x + 2)\text{,}\) so the graph of \(g\) is shifted \(2\) units to the left of \(y = e^x\text{.}\) Also, \(h(x) = f(x) + 2\text{,}\) so the graph of \(h\) is shifted \(2\) units up from \(y = e^x\text{.}\) The graphs are shown above.
Checkpoint 5.41
Use your calculator to evaluate the following powers.
 \(e^2\)
 \(e^{3.5}\)
 \(e^{0.5}\)
Answer
\(e^2\approx 7.389\)
\(e^{3.5}\approx 33.115 \)
\(e^{0.5}\approx 0.6065 \)
Subsection The Natural Logarithmic Function
The base \(e\) logarithm of a number \(x\text{,}\) or \(\log_ e x\text{,}\) is called the natural logarithm of \(x\) and is denoted by \(\ln x\text{.}\)
The Natural Logarithm
The natural logarithm is the logarithm base \(e\text{.}\)
\begin{equation*}
\ln x = \log_{e}{x}, ~~~~ x\gt 0
\end{equation*}
The natural logarithm of \(x\) is the exponent to which \(e\) must be raised to produce \(x\text{.}\) For example, the natural logarithm of \(10\text{,}\) or \(\ln 10\text{,}\) is the solution of the equation
\begin{equation*}
e^y = 10
\end{equation*}
You can verify on your calculator that
\begin{equation*}
e^{\alert{2.3}} ~~ \approx 10 \text{ or } ~~ \ln 10 \approx \alert{2.3}
\end{equation*}
In general, natural logs obey the same conversion formulas that work for logs to other bases.
Conversion Formulas for Natural Logs
\begin{equation*}
\blert{y = \ln x} ~~\text{ if and only if } ~~ \blert{e^y = x}
\end{equation*}
In particular,
\begin{equation*}
\begin{aligned}[t]
\ln e \amp = 1 \text{ because } e^1 = e \\
\ln 1 \amp = 0 \text{ because } e^0 = 1 \\
\end{aligned}
\end{equation*}
The conversion formulas tell us that the natural log function, \(g(x) = \ln x\text{,}\) is the inverse function for the natural exponential function, \(f(x) = e^x\text{.}\)
Example 5.42
 Graph \(f(x) = e^x\) and \(f^{1}(x) = \ln x\) on the same grid.
 Give the domain and range of the natural log function.
Solution

We can make a table of values for \(f^{1}(x) = \ln x\) by interchanging the columns in the table for \(f(x) = e^x\text{.}\) Plotting the points gives us the graph below.
\(x\) 
\(y=\ln x\) 
\(0.050\) 
\(3\) 
\(0.135\) 
\(2\) 
\(0.368\) 
\(1\) 
\(1\) 
\(0\) 
\(2.718\) 
\(1\) 
\(7.389\) 
\(2\) 
\(20.086\) 
\(3\) 
 The domain of the natural log function is the same as the range of \(y = e^x\text{,}\) or all positive numbers. The range of \(y = \ln x\) is the same as the domain of \(y = e^x\text{,}\) or all real numbers. These results are confirmed by the graph of \(y = \ln x\text{.}\)
Checkpoint 5.44
Use your calculator to evaluate each logarithm. Round your answers to four decimal places.
 \(\ln 100\)
 \(\ln 0.01\)
 \(\ln e^3\)
Answer
 \(\ln 100\approx 4.6052\)
 \(\ln 0.01\approx 4.6052\)
 \(\ln e^3 =3\)
Subsection Exponential Growth and Decay
In Section 4.1, we considered functions of the form
\begin{equation*}
P(t) = P_0\cdot b^t
\end{equation*}
which describe exponential growth when \(b \gt 1\) and exponential decay when \(0 \lt b \lt 1\text{.}\) Exponential growth and decay can also be modeled by functions of the form
\begin{equation*}
\blert{P(t) = P_0 \cdot e^{kt}}
\end{equation*}
where we have substituted \(e^k\) for the growth factor \(b\text{,}\) so that
\begin{equation*}
\begin{aligned}[t]
P(t) \amp = P_0 \cdot b^t\\
\amp = P_0 \cdot \left(e^k\right)^t = P_0 \cdot e^{kt}\\
\end{aligned}
\end{equation*}
We can find the value of \(k\) by solving the equation \(b = e^k\) for \(k\text{,}\) to get \(k = \ln b\text{.}\)
For instance, in Example 4.1 in Section 4.1 we found that a colony of bacteria grew according to the formula
\begin{equation*}
P(t) = 100 \cdot \alert{3^t}
\end{equation*}
We can express this function in the form \(P(t) = 100 \cdot \alert{e^{kt}}\) if we set
\begin{equation*}
3 = e^k ~ \text{ or } ~ k = \ln 3 \approx 1.0986
\end{equation*}
Thus, the growth law for the colony of bacteria can be written
\begin{equation*}
P(t) \approx 100 \cdot e^{1.0986t}
\end{equation*}
By graphing both functions on your calculator, you can verify that
\begin{equation*}
P(t) = 100 \cdot 3t~~~\text{and}~~~P(t) = 100 \cdot e^{1.0986t}
\end{equation*}
are just two ways of writing the same function.
Example 5.54
From 1990 to 2000, the population of Clark County, Nevada, grew by \(6.4\%\) per year.
 What was the growth factor for the population of Clark County from 1990 to 2000? If the population of Clark County was \(768,000\) in 1990, write a formula for the population \(t\) years later.
 Write a growth formula for Clark County using base \(e\text{.}\)
Solution

The growth factor was \(b = 1 + r = 1.064\text{.}\) The population \(t\) years later was
\begin{equation*}
P(t) = 768,000 (1.064)^t
\end{equation*}

We use the formula \(P(t) = P_0 \cdot e^{kt}\text{,}\) where \(e^k = 1.064\text{.}\) Solving for \(k\text{,}\) we find
\begin{equation*}
k = \ln 1.064 = 0.062
\end{equation*}
so \(P(t) = 768,000 e^{0.062t}\text{.}\)
Checkpoint 5.55
From 1994 to 1998, the number of personal computers connected to the Internet grew according to the formula \(N(t) = 2.8e^{0.85t}\text{,}\) where \(t = 0\) in 1994 and \(N\) is in millions. (Source: Los Angeles Times, September 6, 1999)
 Evaluate \(N(1)\text{.}\) By what percent did the number of Internet users grow in one year?
 Express the growth law in the form \(N(t) = N_0 (1 + r)^t\text{.}\)
Hint
Answer
\(N(1)\approx 6.55\text{,}\) \(~134\%\)
\(N(t)\approx 2.8 (1.3396)^t \)
If \(k\) is negative, then \(e^k\) is a fraction less than \(1\text{.}\) For example, if \(k = 2\text{,}\)
\begin{equation*}
e^{2} = \frac{1}{e^2} \approx \frac{1}{7.3891} \approx 0.1353
\end{equation*}
Thus, for negative values of \(k\text{,}\) the function \(P(t) = P_0 e^{kt}\) describes exponential decay.
Exponential Growth and Decay
The function
\begin{equation*}
P(t) = P_0 e^{kt}
\end{equation*}
describes exponential growth if \(k \gt 0\text{,}\) and exponential decay if \(k \lt 0\text{.}\)
Example 5.56
Express the decay law \(N(t) = 60 (0.8)^t\) in the form \(N(t) = N_0 e^{kt}\text{.}\)
Solution
For this decay law, \(N_0 = 60\) and \(b = 0.8\text{.}\) We would like to find a value for \(k\) so that \(e^k = b = 0.8\text{,}\) that is, we must solve the equation
\begin{equation*}
\begin{aligned}[t]
e^k \amp = 0.8\amp\amp \blert{\text{Take natural log of both sides.}}\\
\ln e^k\amp = \ln 0.8\amp\amp \blert{\text{Simplify.}}\\
k\amp = \ln 0.8 \approx 0.2231
\end{aligned}
\end{equation*}
Replacing \(b\) with \(e^k\text{,}\) we find that the decay law is
\begin{equation*}
N(t) \approx 60e^{0.2231t}
\end{equation*}
Checkpoint 5.57
A scientist isolates \(25\) grams of krypton91, which decays according to the formula
\begin{equation*}
N(t) = 25e^{0.07t}\text{,}
\end{equation*}
where \(t\) is in seconds.

Complete the table of values showing the amount of krypton91 left at \(10\)second intervals over the first minute.
\(t\) 
\(0\) 
\(10\) 
\(20\) 
\(30\) 
\(40\) 
\(50\) 
\(60\) 
\(N(t)\) 
\(\) 
\(\) 
\(\) 
\(\) 
\(\) 
\(\) 
\(\) 
Use the table to choose a suitable window and graph the function \(N(t)\text{.}\)
Write and solve an equation to answer the question: How long does it take for 60% of the krypton91 to decay?
HintIf \(60\%\) of the krypton91 has decayed, \(40\%\) of the original \(25\) grams remains.
Answer
\(t\) 
\(0\) 
\(10\) 
\(20\) 
\(30\) 
\(40\) 
\(50\) 
\(60\) 
\(N(t)\) 
\(25\) 
\(12.41\) 
\(6.16\) 
\(3.06\) 
\(1.52\) 
\(0.75\) 
\(0.37\) 
\(25 e^{0.07t} = 0.40(25)\text{;}\) \(~~t=\dfrac{\ln(0.4)}{0.07}\approx 13.09 \) seconds
Subsection Homework 5.3
¶
For Problems 14, use your calculator to complete the table for each function. Then choose a suitable window and graph the function.
\(x\) 
\(10\) 
\(5\) 
\(0\) 
\(5\) 
\(10\) 
\(15\) 
\(20\) 
\(f(x)\) 
\(\hphantom{0000}\) 
\(\hphantom{0000}\) 
\(\hphantom{0000}\) 
\(\hphantom{0000}\) 
\(\hphantom{0000}\) 
\(\hphantom{0000}\) 
\(\hphantom{0000}\) 
1
\(f(x)=e^{0.2x} \)
2
\(f(x)=e^{0.6x} \)
3
\(f(x)= e^{0.3x} \)
4
\(f(x)=e^{0.1x} \)
For Problems 56, simplify.
5
\(\ln e^2\)
\(e^{\ln 5t} \)
\(e^{\ln x} \)
\(\ln \sqrt{e} \)
6
\(\ln e^{x^4}\)
\(e^{3 \ln x} \)
\(e^{\ln x \ln y} \)
\(\ln \left(\dfrac{1}{e^{2t}} \right) \)
For Problems 710, solve for \(x\text{.}\) Round your answers to two decimal places.
7
\(e^x=1.9\)
\(e^x=45\)
\(e^x=0.3\)
8
\(e^x=2.1\)
\(e^x=60\)
\(e^x=0.9\)
9
\(\ln x=1.42\)
\(\ln x = 0.63\)
\(\ln x = 2.6\)
10
\(\ln x=2.03\)
\(\ln x = 0.59\)
\(\ln x = 3.4\)
11
The number of bacteria in a culture grows according to the function
\begin{equation*}
N(t) = N_0 e^{0.04t}
\end{equation*}
where \(N_0\) is the number of bacteria present at time \(t = 0\) and \(t\) is the time in hours.
Write a growth law for a sample in which \(6000\) bacteria were present initially.
Make a table of values for \(N(t)\) in \(5\)hour intervals over the first \(30\) hours.
Graph \(N(t) \text{.}\)
How many bacteria were present at \(t = 24\) hours?
How much time must elapse (to the nearest tenth of an hour) for the original \(6000\) bacteria to increase to \(100,000\text{?}\)
12
Hope invests \(\$2000\) in a savings account that pays \(5\frac{1}{2}\%\) annual interest compounded continuously.
Write a formula that gives the amount of money \(A(t)\) in Hope’s account after \(t\) years.
Make a table of values for \(A(t)\) in \(2\)year intervals over the first \(10\) years.
Graph \(A(t) \text{.}\)
How much will Hope's account be worth after \(7\) years?
How long will it take for the account to grow to \(\$5000\text{?}\)
13
The intensity, \(I\) (in lumens), of a light beam after passing through \(t\) centimeters of a filter having an absorption coefficient of \(0.1\) is given by the function
\begin{equation*}
I (t) = 1000e^{0.1t}
\end{equation*}
Graph \(I (t)\text{.}\)
What is the intensity (to the nearest tenth of a lumen) of a light beam that has passed through \(0.6\) centimeter of the filter?
How many centimeters (to the nearest tenth) of the filter will reduce the illumination to \(800\) lumens?
14
Xrays can be absorbed by a lead plate so that
\begin{equation*}
I (t) = I_0 e^{1.88t}
\end{equation*}
where \(I_0\) is the Xray count at the source and \(I (t)\) is the Xray count behind a lead plate of thickness \(t\) inches.
Graph \(I (t)\text{.}\)
What percent of an Xray beam will penetrate a lead plate \(\frac{1}{2}\) inch thick?
How thick should the lead plate be in order to screen out \(70\%\) of the Xrays?
For problems 1518, express each exponential function in the form \(P(t) = P_0b^t\text{.}\) Is the function increasing or decreasing? What is itsinitial value?
15
\(P(t) = 20e^{0.4t}\)
16
\(P(t)=0.8 e^{1.3t} \)
17
\(P(t) = 6500e^{2.5t}\)
18
\(P(t)=1.7 e^{0.02t} \)
19

Fill in the table, rounding your answers to four decimal places.
\(x\) 
\(0\) 
\(0.5\) 
\(1\) 
\(1.5\) 
\(2\) 
\(2.5\) 
\(e^x\) 
\(\phantom{000} \) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
Compute the ratio of each function value to the previous one. Explain the result.
20

Fill in the table, rounding your answers to four decimal places.
\(x\) 
\(0\) 
\(2\) 
\(4\) 
\(6\) 
\(8\) 
\(10\) 
\(e^x\) 
\(\phantom{000} \) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
Compute the ratio of each function value to the previous one. Explain the result.
21

Fill in the table, rounding your answers to the nearest integer.
\(x\) 
\(0\) 
\(0.6931\) 
\(1.3863\) 
\(2.0794\) 
\(2.7726\) 
\(3.4657\) 
\(4.1589\) 
\(e^x\) 
\(\phantom{000} \) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
Subtract each \(x\)value from the next one. Explain the result.
22

Fill in the table, rounding your answers to the nearest integer.
\(x\) 
\(0\) 
\(1.0986\) 
\(2.1972\) 
\(3.2958\) 
\(4.3944\) 
\(5.4931\) 
\(6.5917\) 
\(e^x\) 
\(\phantom{000} \) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
\(\phantom{000}\) 
Subtract each \(x\)value from the next one. Explain the result.
For Problems 23–30, solve. Round your answers to two decimal places.
23
\(6.21 = 2.3e^{1.2x}\)
24
\(22.26 = 5.3e^{0.4x}\)
25
\(6.4 = 20e^{0.3x}  1.8 \)
26
\(4.5 = 4e^{2.1x} + 3.3 \)
27
\(46.52 = 3.1e^{1.2x} + 24.2 \)
28
\(1.23 = 1.3e^{2.1x}  17.1 \)
29
\(16.24 = 0.7e^{1.3x}  21.7 \)
30
\(55.68 = 0.6e^{0.7x} + 23.1 \)
For Problems 3136, solve the equation for the specified variable.
31
\(y = e^{kt},~~\) for \(t\)
32
\(\dfrac{T}{R} = e^{t/2},~~\) for \(t\)
33
\(y = k(1e^{t}),~~\) for \(t\)
34
\(B  2 = (A + 3)e^{t/3},~~\) for \(t\)
35
\(T = T_0 \ln(k + 10),~~\) for \(k\)
36
\(P = P_0 + \ln 10k,~~\) for \(k\)
37

Fill in the table, rounding your answers to three decimal places.
\(n\) 
\(0.39\) 
\(3.9\) 
\(39\) 
\(390\) 
\(\ln n\) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
Subtract each natural logarithm in your table from the next one. (For example, compute \(\ln 3.9  \ln 0.39\text{.}\)) Explain the result.
38

Fill in the table, rounding your answers to three decimal places.
\(n\) 
\(0.64\) 
\(6.4\) 
\(64\) 
\(640\) 
\(\ln n\) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
Subtract each natural logarithm in your table from the next one. (For example, compute \(\ln 6.4  \ln 0.64\text{.}\)) Explain the result.
39

Fill in the table, rounding your answers to three decimal places.
\(n\) 
\(2\) 
\(4\) 
\(8\) 
\(16\) 
\(\ln n\) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
Divide each natural logarithm in your table by \(\ln 2\text{.}\) Explain the result.
40

Fill in the table, rounding your answers to three decimal places.
\(n\) 
\(5\) 
\(25\) 
\(125\) 
\(625\) 
\(\ln n\) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
\(\hphantom{0000} \) 
Divide each natural logarithm in your table by \(\ln 5\text{.}\) Explain the result.
For Problems 41–46,
Express each growth or decay law in the form \(N(t) = N_0e^{kt}\text{.}\)
Check your answer by graphing both forms of the function on the same axes. Do they have the same graph?
41
\(N(t) = 100\cdot 2^t\)
42
\(N(t) = 50\cdot 3^t\)
43
\(N(t) = 1200(0.6)^t\)
44
\(N(t) = 300(0.8)^t\)
45
\(N(t) = 10(1.15)^t\)
46
\(N(t) = 1000(1.04)^t\)
47
The population of Citrus Valley was \(20,000\) in \(2000\text{.}\) In \(2010\text{,}\) it was \(35,000\text{.}\)
What is \(P_0\) if \(t = 0\) in \(2000\text{?}\)
Use the population in \(2010\) to find the growth factor \(e^k\text{.}\)
Write a growth law of the form \(P(t) = P_0 e^{kt}\) for the population of Citrus Valley.
If it continues at the same rate of growth, what will the population be in \(2030\text{?}\)
48
A copy of Time magazine cost $\(1.50\) in \(1981.~\) In \(1988\text{,}\) the cover price had increased to $\(2.00\text{.}\)
What is \(P_0\) if \(t = 0\) in \(1981\text{?}\)
Use the price in \(1988\) to find the growth factor \(e^k\text{.}\)
Find a growth law of the form \(P(t) = P_0e^{kt}\) for the price of Time.
In \(1999\text{,}\) a copy of Time cost $\(3.50\text{.}\) Did the price of the magazine continue to grow at the same rate from \(1981\) to \(1999\text{?}\)
49
Cobalt60 is a radioactive isotope used in the treatment of cancer. A \(500\)milligram sample of cobalt60 decays to \(385\) milligrams after \(2\) years.
Using \(P_0 = 500\text{,}\) find the decay factor \(e^k\) for cobalt60.
Write a decay law \(N(t) = N_0e^{kt}\) for cobalt60.
How much of the original sample will be left after \(10\) years?
50
Weed seeds can survive for a number of years in the soil. An experiment on cultivated land found \(155\) million weed seeds per acre, and in the following years the experimenters prevented the seeds from coming to maturity and producing new weeds. Four years later, there were \(13.6\) million seeds per acre. (Source: Burton, 1998)
Find the annual decay factor \(e^k\) for the number of weed seeds in the soil.
Write an exponential formula with base \(e\) for the number of weed seeds that survived after \(t\) years.
Problems 51–58 are about doubling time and halflife.
51
Delbert invests $\(500\) in an account that pays \(9.5\%\) interest compounded continuously.
Write a formula for \(A(t)\) that gives the amount of money in Delbert's account after \(t\) years.
How long will it take Delbert's investment to double to $\(1000\text{?}\)
How long will it take Delbert's money to double again, to $\(2000\text{?}\)
Graph \(A(t)\) and illustrate the doubling time on your graph.
Choose any point \((t_1, A_1)\) on the graph, then find the point on the graph with vertical coordinate \(2A_1\text{.}\) Verify that the difference in the \(t\)coordinates of the two points is the doubling time.
52
The growth of plant populations can be measured by the amount of pollen they produce. The pollen from a population of pine trees that lived more than \(9500\) years ago in Norfolk, England, was deposited in the layers of sediment in a lake basin and dated with radiocarbon techniques.
The figure shows the rate of pollen accumulation plotted against time, and the fitted curve \(P(t) = 650e^{0.00932t}\text{.}\) (Source: Burton, 1998)
What was the annual rate of growth in pollen accumulation?
Find the doubling time for the pollen accumulation, that is, the time it took for the accumulation rate to double.
By what factor did the pollen accumulation rate increase over a period of \(500\) years?
53
Technetium99m (Tc99m) is an artificially produced radionuclide used as a tracer for producing images of internal organs such as the heart, liver, and thyroid. A solution of Tc99m with initial radioactivity of \(10,000\) becquerels (Bq) decays according to the formula
\begin{equation*}
N(t) = 10,000e^{0.1155t}
\end{equation*}
where \(t\) is in hours.
How long will it take the radioactivity to fall to half its initial value, or \(5000\) Bq?
How long will it take the radioactivity to be halved again?
Graph \(N(t)\) and illustrate the halflife on your graph.
Choose any point \((t_1, N_1)\) on the graph, then find the point on the graph with vertical coordinate \(0.5N_1\text{.}\) Verify that the difference in the \(t\)coordinates of the two points is the halflife.
54
All living things contain a certain amount of the isotope carbon14. When an organism dies, the carbon14 decays according to the formula
\begin{equation*}
N(t) = N_0e^{0.000124t}
\end{equation*}
where \(t\) is measured in years. Scientists can estimate the age of an organic object by measuring the amount of carbon14 remaining.
When the Dead Sea scrolls were discovered in 1947, they had \(78.8\%\) of their original carbon14. How old were the Dead Sea scrolls then?
What is the halflife of carbon14, that is, how long does it take for half of an object's carbon14 to decay?
55
The halflife of iodine131 is approximately \(8\) days.
If a sample initially contains \(N_0\) grams of iodine131, how much will it contain after \(8\) days? How much will it contain after \(16\) days? After \(32\) days?
Use your answers to part (a) to sketch a graph of \(N(t)\text{,}\) the amount of iodine131 remaining, versus time. (Choose an arbitrary height for \(N_0\) on the vertical axis.)
Calculate \(k\text{,}\) and hence find a decay law of the form \(N(t) = N_0e^{kt}\text{,}\) where \(k \lt 0\text{,}\) for iodine131.
56
The halflife of hydrogen3 is \(12.5\) years.
If a sample initially contains \(N_0\) grams of hydrogen3, how much will it contain after \(12.5\) years? How much will it contain after \(25\) years?
Use your answers to part (a) to sketch a graph of \(N(t)\text{,}\) the amount of hydrogen3 remaining, versus time. (Choose an arbitrary height for \(N_0\) on the vertical axis.)
Calculate \(k\text{,}\) and hence find a decay law of the form \(N(t) = N_0e^{kt}\text{,}\) where \(k\lt 0\text{,}\) for hydrogen3.
57
A Geiger counter measures the amount of radioactive material present in a substance. The table shows the count rate for a sample of iodine128 as a function of time. (Source: Hunt and Sykes, 1984)
Time (min) 
\(0\) 
\(10\) 
\(20\) 
\(30\) 
\(40\) 
\(50\) 
\(60\) 
\(70\) 
\(80\) 
\(90\) 
Counts/sec 
\(120\) 
\(90\) 
\(69\) 
\(54\) 
\(42\) 
\(33\) 
\(25\) 
\(19\) 
\(15\) 
\(13\) 
Graph the data and use your calculator's exponential regression feature to fit a curve to them.
Write your equation in the form \(G(t) = G_0e^{kt}\text{.}\)
Calculate the halflife of iodine128.
58
The table shows the count rate for sodium24 registered by a Geiger counter as a function of time. (Source: Hunt and Sykes, 1984)
Time (min) 
\(0\) 
\(10\) 
\(20\) 
\(30\) 
\(40\) 
\(50\) 
\(60\) 
\(70\) 
\(80\) 
\(90\) 
Counts/sec 
\(180\) 
\(112\) 
\(71\) 
\(45\) 
\(28\) 
\(18\) 
\(11\) 
\(7\) 
\(4\) 
\(3\) 
Graph the data and use your calculator's exponential regression feature to fit a curve to them.
Write your equation in the form \(G(t) = G_0e^{kt}\text{.}\)
Calculate the halflife of sodium24.