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Math Topics

Precalculus

Exponential and Logarithmic Functions

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1
Inverse Functions

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

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.

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

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.

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

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2
Exponential Function

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.

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

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

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

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

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 \(6\).

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

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

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

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

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

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

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

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

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 \(\text{pH}=-\log{\left[\text{H}^+\right]}\).

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

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

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

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.

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

Find the value of \(k\).

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

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

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.

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 \(t\), \(t\geq 0\).

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

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

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.

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

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

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

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.

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.

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

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

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

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

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

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.

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?