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College Algebra
Exponential and Logarithmic Functions

1
Inverse Functions

Problem  1

Verify that the following functions are inverses of each other
\(f(x)=3x-4\);
\(g(x)=\dfrac{x}{3}+\dfrac{4}{3}\)

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Problem  2
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Find the inverse of the function \(f(x)=-2x+3\) and check that it is the inverse.

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Problem  3

Which of the following functions are one-to-one and therefore have an inverse?

  1. \(f(t)=-3t+1\)

  2. The function \(f\) given graphically in Figure 4.

  3. The function \(f\) given by the values in Table 2.

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Problem  4
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Graph the function \(f(x)=-2x+3\) and its inverse, \(f^{-1}(x)=-\dfrac{1}{2}x+\dfrac{3}{2}\), on the same set of axes, using the same scale for both axes. What do you observe?

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Problem  5

Let \(f(x)=x^3+1\)

  1. Show that \(f\) is one-to-one using the horizontal line test.

  2. Find the inverse of \(f\).

  3. Graph \(f\) and its inverse on the same set of axes.

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Problem  6

Show that the function \(g(x)=x^2\), \(x\geq 0\) has an inverse and find it.

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Problem  7

Table 4 lists certain quantities of fuel in gallons and the corresponding quantities in liters.

  1. There are \(3.785\) liters in \(1\) gallon. Find an expression for a function \(L(x)\) which will take the number of gallons of fuel as its input and give the number of liters of fuel as its output.

  2. Rewrite Table 4 so that the number of liters is the input and the number of gallons is the output.

  3. Find an expression for a function \(G(x)\) that will take the number of liters of fuel as its input and give the number of gallons of fuel as its output.

  4. Show that \(L(x)\) and \(G(x)\) are inverses of each other.

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2
Exponential Function

Problem  1

Evaluate each expression to four decimal places using a calculator.

  1. \(0.5^{\frac{1}{3}}\)

  2. \(2.8^{2.5}\)

  3. \(4^\sqrt{2}\)

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Problem  2

Make a table of values of the exponential function \(f(x)=2^x\). Use the table to sketch the graph of the function. What happens to the value of the function as \(x\to\pm\infty\)?

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Problem  3

Make a table of values of the exponential function \(f(x)=5\left(\dfrac{1}{3}\right)^x=5\left(3^{-x}\right)\). Use the table to sketch the graph of the function. What happens to the value of the function as \(x\to\pm\infty\)? Determine the range of the function from the graph.

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Problem  4

Make a table of values of the function \(h(x)=-(2)^{2x}\) and sketch the graph of the function. Find the \(y\)-intercept, domain, and range. Describe the behavior of the function as \(x\) approaches \(\pm\infty\).

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Problem  5

Use the graph of \(f(x)=3^x\) to sketch the graph of each function.

  1. \(g(x)=3^{x-2}\)

  2. \(h(x)=3^x-1\)

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Problem  6

Evaluate each expression to four decimal places using a calculator.

  1. \(e^\frac{1}{3}\)

  2. \(e^{2.5}\)

  3. \(e^{-2}\)

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Problem  7

Make a table of values for the function \(g(x)=e^x+3\) and sketch the graph of the function. Find the \(y\)-intercept, domain, and range. Describe the behavior of the function as \(x\) approaches \(\pm\infty\).

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Problem  8
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Suppose \(\$2500\) is invested in a savings account. Find the following quantities.

  1. Amount in the account after \(4\) years if the interest rate is \(5.5\%\) compounded monthly.

  2. Amount in the account after \(4\) years if the interest rate is \(5.5\%\) compounded continuously.

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Problem  9

A Honda Civic (2-Door coupe) depreciates at a rate of about \(8\%\) per year. This means that each year it will lose \(8\%\) of the value it had the previous year. If the Honda Civic was purchased at \(\$20\text{,}000\), make a table of its value over the first \(5\) years after purchase. Find a function that gives its value \(t\) years after purchase, and sketch its graph.

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3
Logarithmic Functions

Problem  1

Complete the table by filling in the exponential statements that are equivalent to the given logarithmic statements.

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Problem  2
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Complete the following table by filling in the logarithmic statements that are equivalent to the given exponential statements.

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Problem  3
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Evaluate the following without using a calculator. If there is no solution, so state.

  1. \(\log_5{125}\)

  2. \(\log_{10}{\frac{1}{100}}\)

  3. \(\log_a{a^4}\), \(a>0\)

  4. \(3^{\log_3{5}}\)

  5. \(\log_{10}{-1}\)

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Problem  4

Use the definition of the logarithm to find the value of \(x\).

  1. \(\log_4{16}=x\)

  2. \(\log_3{x}=-2\)

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Problem  5

Without using a calculator, evaluate the following expressions.

  1. \(\log{10\text{,}000}\)

  2. \(\ln{e^\frac{1}{2}}\)

  3. \(e^{\ln{a}}\), \(a>0\)

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Problem  6
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Use the change-of-base formula with the indicated logarithm to calculate the following:

  1. \(\log_6{15}\), using common logarithm

  2. \(\log_7{0.3}\), using natural logarithm

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Problem  7

Graph \(f(x)=\log_2 x\) by first making a table of values.

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Problem  8
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Find the domain of each of the following logarithmic functions. Sketch a graph of the function and find its range. Indicate the vertical asymptote.

  1. \(g(t)=\ln {(-t)}\)

  2. \(f(t)=3\log_2{(x-1)}\)

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Problem  9

If \(T(x)=-5.7\ln x+34.22\), where \(x\) is the speed of the wind, in miles per hour, and \(T(x)\) is the wind chill temperature, then

  1. Find the wind chill temperature when the wind speed is \(20\) miles per hour, and the temperature with no wind is \(30^\circ\text{F}\). Round your answer to the nearest tenth.

  2. Use a graphing utility to sketch a graph of \(T(x)\) and explain what happens as the wind speed increases.

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Problem  10

Since the intensities of earthquakes vary widely, they are measured on a logarithmic scale known as the Richter Scale, using the formula \(R(I)=\log \left(\dfrac{I}{I_0}\right)\) where \(I\) represents the actual intensity of the earthquake, and \(I_0\) is a baseline intensity used for comparison. The Richter Scale gives the magnitude of the earthquake. Because of the logarithmic nature of this function, an increase of a single unit in the value of \(R(I)\) represents a tenfold increase in the intensity of the earthquake. A recording of \(7\), for example, corresponds to an intensity that is \(10\) times as large as the intensity of an earthquake with a recording of \(6\).

  1. If the intensity of an earthquake is \(100\) times the baseline intensity \(I_0\), what is the magnitude on the Richter Scale?

  2. A 2003 earthquake in San Simeon, CA registered \(6.5\) on the Richter Scale. Express its intensity in terms of \(I_0\).

  3. A 2004 earthquake in central Japan registered \(5.4\) on the Richter Scale. Express its intensity in terms of \(I_0\). What is the ratio of the intensity of the 2003 San Simeon quake to the intensity os this quake?

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4
Properties of Logarithms

Problem  1

Given that \(\log{2.5}\approx 0.3979\) and \(\log 3\approx 0.4771\), first calculate the following logarithms without the us of a calculator. Then check your answers with a calculator.

  1. \(\log {25}\)

  2. \(\log {75}\)

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Problem  2
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Simplify the following expressions, if possible, by eliminating exponents and radicals. Assume \(x\), \(y>0\).

  1. \(\log{\left(xy^{-3}\right)}\)

  2. \(\log{\left(3x^\frac{1}{2}\sqrt[3]{y}\right)}\)

  3. \(\left(\ln x\right)^\frac{1}{3}\)

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Problem  3
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Write the following as a sum and/or difference of logarithmic expressions. Eliminate exponents and radicals wherever possible.

  1. \(\log{\left(\dfrac{x^3y^2}{100}\right)}, x, y>0\)

  2. \(\log{\left(\dfrac{\sqrt{x^2+2}}{(x+1)^3}\right)}, x>-1\)

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Problem  4
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Write each of the following as the logarithm of a single quantity.

  1. \(\log_a{3}+\log_a{6}\), \(a>0\)

  2. \(\dfrac{1}{3}\ln{64}+\dfrac{1}{2}\ln{x}\), \(x>0\)

  3. \(3\log{5}-1\)

  4. \(\log_a{x}+\dfrac{1}{2}\log_a\left(x^2+1\right)-\log_a{3}\), \(a>0\), \(x>0\)

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Problem  5
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The pH of a solution is a measure of the concentration of hydrogen ions in the solution. This concentration, which is denoted by \(\left[\text{H}^+\right]\), is given in units pf moles per liter, where one mole is \(6.02\times 10^{23}\) molecules. Because the concentration of hydrogen ions can vary by several powers of \(10\) from one solution to another, the pH scale was introduced to express the concentration in more accessible terms. The pH of a solution is defined as \(\text{pH}=-\log{\left[\text{H}^+\right]}\).

  1. Find the pH of solution \(A\), whose hydrogen ion concentration is \(10^{-4}\) moles/liter.

  2. Find the pH of solution \(B\), whose hydrogen ion concentration is \(4.1\times 10^{-8}\) moles/liter.

  3. If a solution has a pH of \(9.2\), what is its concentration of hydrogen ions?

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5
Exponential and Logarithmic Equations

Problem  1
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Solve the equation \(2^t=128\).

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Problem  2

Solve the equation \(10^{2x-1}=3^x\)

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Problem  3
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Solve the equation \(3e^{2t}+6=24\)

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Problem  4

Solve the equation \(e^{2x}-2e^x-3=0\)

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Problem  5
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Suppose a bank pays interest at a rate of \(5\%\), compounded continuously, on an initial deposit of \(\$1000\) to grow to a total of \(\$1200\), assuming that no withdrawals or additional deposits are made?

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Problem  6

Suppose a colony of bacteria doubles its initial population of \(10\text{,}000\) in \(10\) hours. Assume the function that models this growth is given by \(P(t)=P_0e^{kt}\), where \(t\) is given in hours, and \(P_0\) is the initial population.

  1. Find the population at time \(t=0\).

  2. Find the value of \(k\).

  3. What is the population at time \(t=20\)?

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Problem  7

Solve the equation \(4+\log_3 x=6\)

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Problem  8

Solve the equation \(\log 2x+ \log(x+4)=1\)

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Problem  9
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Solve the equation \(\ln x=2+\ln(x-1)\)

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Problem  10

The cumulative box office revenue from the movie Finding Nemo can be modeled by the logarithmic function \(R(x)=78.046\ln(x+1)+114.36\) where \(x\) is the number of weeks since the movie opened and \(R(x)\) is given in millions of dollars. How many weeks after the opening of the movie was the cumulative revenue equal to \(\$300\) million?

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6
Exponential, Logistic, and Logarithmic Models

Problem  1
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It takes \(29\) years for an initial amount \(A_0\) of strontium-90 to break down into half the original amount \(\dfrac{A_0}{2}\). That is, the half-life of strontium-90 is \(29\) years.

  1. Given an initial amount of \(A_0\) grams of strontium-90, at \(t=0\), find an exponential decay model, \(A(t)=A_0e^{kt}\) that gives the amount of strontium-90 at time \(t\), \(t\geq 0\).

  2. Calculate the time required for strontium-90 to decay to \(\dfrac{1}{10}A_0\).

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Problem  2

The population of the United States is expected to grow from \(282\) million in 2000 to \(335\) million in 2020.

  1. Find a function of the form \(P(t)=Ce^{kt}\) that which models the population growth. Here, \(t\), is the number of years after 2000, and \(P(t)\) is in millions.

  2. Use your model to predict the population of the United States in 2016.

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Problem  3

Table 1 shows the United States national debt (in billions of dollars) for selected years from 1975-2005.

  1. Make a scatter plot of the data and find an exponential function of the form \(f(x)=Ca^x\) that best fits the data.

  2. From the model in part (a) what was the estimated national debt in the year 2010? Compare your answer with the actual value of \(\$13\text{,}000\) billion dollars.

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Problem  4

When designing buildings, engineers must pay careful attention to how different factors affect the load a structure can carry. Table 2 gives the load, in pounds of concrete, when a \(1\)-inch-diameter anchor is used as a joint. The table summarizes the relation between the load and how deep the anchor is drilled into the concrete.

  1. From examining the table, what is the general relationship between the depth of the anchor and the load?

  2. Make a scatterplot of the data and find a natural logarithmic function that best fits the data.

  3. If an anchor were drilled \(10\) inches deep, what is the resulting load that can be carried?

  4. What is the maximum depth an anchor should be drilled in order to sustain a load of \(9000\) pounds?

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Problem  5

Table 4 gives the population os South America for selected years from 1970 to 2000.

  1. Use a graphing utility to make a scatter plot of the data and find the logistic function of the form \(f(x)=\dfrac{c}{1+ae^{-bx}}\) that best fits the data from 1970 to 2000. Let \(x\) be the number of years from 1970.

  2. From this model, what is the projected population in 2020? How does it compare with the projection of \(421\) million given by the U.S. Census Bureau?

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7
Test & Summary

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