Exponential and logarithmic functions | Practice Problems, Examples & Solutions

Exponential Functions and Their Graphs

Introductory and Intermediate Algebra for College Students

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function $f(x)=0.48 \ln (x+1)+27$ models the barometric air pressure, $f(x),$ in inches of mercury, at a distance of $x$ miles from the eye of a hurricane. Use this function to solve.
The function $P(t)=145 e^{-0.092 t}$ models a runner's pulse,
$P(t),$ in beats per minute, $t$ minutes after a race, where $0 \leq t \leq 15 .$ Graph the function using a graphing utility. TRACE along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

Exponential and Logarithmic Functions

Exponential and Logarithmic Equations

01:09

Introductory and Intermediate Algebra for College Students

use your graphing utility to graph each
side of the equation in the same viewing rectangle. Then use the $x$ -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation.
$2^{x+1}=8$

Exponential and Logarithmic Functions

Exponential and Logarithmic Equations

00:53

Introductory and Intermediate Algebra for College Students

What is a logarithmic equation?

Exponential and Logarithmic Functions

Exponential and Logarithmic Equations

Logarithmic Functions and Their Graphs

886 Practice Problems

01:13

Introductory and Intermediate Algebra for College Students

Solve each equation in Exercises $144-146 .$ Check each proposed solution by direct substitution or with a graphing utility.
$(\ln x)^{2}=\ln x^{2}$

Exponential and Logarithmic Functions

Exponential and Logarithmic Equations

01:33

Introductory and Intermediate Algebra for College Students

Exponential and Logarithmic Functions

Exponential and Logarithmic Equations

00:47

Introductory and Intermediate Algebra for College Students

Explain the differences between solving $\log _{3}(x-1)=4$ and $\log _{3}(x-1)=\log _{3} 4$

Exponential and Logarithmic Functions

Exponential and Logarithmic Equations

Properties of Logarithms

585 Practice Problems

00:33

Introductory and Intermediate Algebra for College Students

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is $1 .$ Where possible, evaluate logarithmic expressions.
$$\log x+3 \log y$$

Exponential and Logarithmic Functions

Properties of Logarithms

01:03

Introductory and Intermediate Algebra for College Students

Make Sense? Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
I expanded $\log _{4} \sqrt{\frac{x}{y}}$ by writing the radical using a rational exponent and then applying the quotient rule, obtaining $\frac{1}{2} \log _{4} x-\log _{4} y.$

Exponential and Logarithmic Functions

Properties of Logarithms

00:52

Introductory and Intermediate Algebra for College Students

Disprove each statement in Exercises $106-110$ by
a. letting y equal a positive constant of your choice, and
b. using a graphing utility to graph the function on each side of the equal sign. The two functions should have different graphs, showing that the equation is not true in general
$$\ln (x y)=(\ln x)(\ln y)$$

Exponential and Logarithmic Functions

Properties of Logarithms

Exponential and Logarithmic Models

230 Practice Problems

00:28

Elementary and Intermediate Algebra 5th

The graphs of $f(x)=\ln x, g(x)=e^{x},$ and $y=x$ are shown in figure (b) below. Describe the relationship between the graphs in words.

Exponential and Logarithmic Functions

Base-e Exponential and Logarithmic Functions

00:58

Elementary and Intermediate Algebra 5th

Explain why the graph of $y=e^{x}-5$ is five units below the graph of $y=e^{x}$.

Exponential and Logarithmic Functions

Base-e Exponential and Logarithmic Functions

02:28

Elementary and Intermediate Algebra 5th

The function $H(s)=-47.73+107.38 \ln s$ approximates the heart rate (in beats/minute) for an Olympic-class cross country skier traveling at $s$ miles per hour, where $s>5$ mph. Find the heart rate of a skier traveling at a rate of 7.5 miles per hour. (Source: btc.ontana.edu/Olympics/physiology)

Exponential and Logarithmic Functions

Base-e Exponential and Logarithmic Functions

Using Exponential and Logarithmic Models to Represent Data

99 Practice Problems

01:43

Introductory and Intermediate Algebra for College Students

Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
In $2006,$ Canada's population exceeded Uganda's by
4.9 million.

Exponential and Logarithmic Functions

Exponential Growth and Decay; Modeling Data

01:41

Introductory and Intermediate Algebra for College Students

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning.
After 100 years, a population whose growth rate is $3 \%$ will have three times as many people as a population whose growth rate is $1 \%$

Exponential and Logarithmic Functions

Exponential Growth and Decay; Modeling Data

04:03

Introductory and Intermediate Algebra for College Students

We used two data points and an exponential function to model the population of the United States from 1970 through 2009. The data are shown again in the table. Use all five data points.
$$\begin{array}{c|c}
\hline x, \text { Number of Years after } 1969 & y, \text { U.S. Population (millions) } \\
\hline 1(1970) & 203.3 \\
\hline 11(1980) & 226.5 \\
\hline 21(1990) & 248.7 \\
\hline 31(2000) & 281.4 \\
\hline 40(2009) & 307.0 \\
\hline
\end{array}$$
Use the values of $r$ in Exercises $45-48$ to select the two models of best fit. Use each of these models to predict by which year the U.S. population will reach 352 million. How do these answers compare to the year we found in Example $1,$ namely $2020 ?$ If you obtained different years, how do you account for this difference?

Exponential and Logarithmic Functions

Exponential Growth and Decay; Modeling Data

Scientific Notation

36 Practice Problems

01:55

Algebra and Trigonometry Real Mathematics, Real People

Each graphing utility screen below shows a model that fits the set of data. The equations and $r^{2}$ -values for the models are given. Determine the equation and coefficient of determination that represents each graph. Explain how you found your result.
(i) $\begin{array}{l}y=1.05 e^{0.02288 x} \\y=1.06 \ln x+0.96\end{array}$
(ii) $\begin{array}{l}r^{2}=0.968 \\r^{2}=0.9991\end{array}$

Exponential and Logarithmic Functions

Nonlinear Models

00:49

College Algebra

Write each expression with a single base.
$$\frac{x^{m} x^{-2}}{x^{-3}}$$

A Review of Basic Algebra

Integer Exponents and Scientific Notation

02:49

College Algebra

Use scientific notation to compute each answer. Write all -answers in scientific notation.
Speed of sound The speed of sound in air is $3.31 \times 10^{4}$ centimeters per second. Compute the speed of sound in meters per minute.

A Review of Basic Algebra

Integer Exponents and Scientific Notation

Algebra and Composition of Functions

8 Practice Problems

02:22

Algebra and Trigonometry Real Mathematics, Real People

Write an equation of the parabola in standard form.

Exponential and Logarithmic Functions

Nonlinear Models

04:41

Algebra and Trigonometry Real Mathematics, Real People

The populations $P$ (in thousands) of Luxembourg for the years 1999 through 2013 are shown in the table, where $t$ represents the year, with $t=9$ corresponding to 1999. (Source: European Commission Eurostat)
$$\begin{array}{|c|c|}\hline \text { Year } & \text { Population, } P \\\hline 1999 & 427.4 \\2000 & 433.6 \\2001 & 439.0 \\2002 & 444.1 \\2003 & 448.3 \\2004 & 455.0 \\2005 & 461.2 \\2006 & 469.1 \\2007 & 476.2 \\2008 & 483.8 \\2009 & 493.5 \\2010 & 502.1 \\2011 & 511.8 \\2012 & 524.9 \\2013 & 537.0 \\\hline\end{array}$$
(a) Use the regression feature of a graphing utility to find a linear model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window.
(b) Use the regression feature of the graphing utility to find a power model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window.
(c) Use the regression feature of the graphing utility to find an exponential model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window.
(d) Use the regression feature of the graphing utility to find a logarithmic model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window.
(e) Which model is the best fit for the data? Explain.
(f) Use each model to predict the populations of Luxembourg for the years 2014 through 2018.
(g) Which model is the best choice for predicting the future population of Luxembourg? Explain.
(h) Were your choices of models the same for parts
(e) and $(g) ?$ If not, explain why your choices were different.

Exponential and Logarithmic Functions

Nonlinear Models

01:36

Algebra and Trigonometry Real Mathematics, Real People

Use the regression feature of a graphing utility to find a power model $y=a x^{b}$ for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window.
$$(1,10.0),(2,4.0),(3,0.7),(4,0.1)$$

Exponential and Logarithmic Functions

Nonlinear Models

Base-e Exponential and Logarithmic Functions

11 Practice Problems

00:24

Algebra and Trigonometry Real Mathematics, Real People

Determine whether the statement is true or false. Justify your answer.
The exponential model $y=a e^{b x}$ represents a growth model when $b>0.$

Exponential and Logarithmic Functions

Nonlinear Models

01:08

Algebra and Trigonometry Real Mathematics, Real People

Use a graphing utility to create a scatter plot of the data. Decide whether the data could best be modeled by a linear model, an exponential model, or a logarithmic model.
$$(1,7.5),(1.5,7.0),(2,6.8),(4,5.0),(6,3.5),(8,2.0)$$

Exponential and Logarithmic Functions

Nonlinear Models

00:54

Algebra and Trigonometry Real Mathematics, Real People

A power model has the form _______ .

Exponential and Logarithmic Functions

Nonlinear Models

Exponential and Logarithmic Equations

24 Practice Problems

01:06

Algebra and Trigonometry Real Mathematics, Real People

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility.
$$2 x^{2} e^{2 x}+2 x e^{2 x}=0$$

Exponential and Logarithmic Functions

Solving Exponential and Logarithmic Equations

02:01

Algebra and Trigonometry Real Mathematics, Real People

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer.
$$\left(16+\frac{0.878}{26}\right)^{3 t}=30$$

Exponential and Logarithmic Functions

Solving Exponential and Logarithmic Equations

00:25

Algebra and Trigonometry Real Mathematics, Real People

Solve the exponential equation.
$$7^{x}=\frac{1}{49}$$

Exponential and Logarithmic Functions

Solving Exponential and Logarithmic Equations

Exponential and Logarithmic Equations and Inequalities

13 Practice Problems

00:59

A Graphical Approach to College Algebra

surface of the ocean due to rapid evaporation. In the higher latitudes, there is less evaporation, and rainfall causes the salinity to be less on the surface than at lower depths. The function given by
$$
f(x)=31.5+1.1 \log (x+1)
$$
models salinity to depths of 1000 meters at a latitude of $57.5^{\circ} \mathrm{N} .$ The variable $x$ is the depth in meters, and $f(x)$ is in grams of salt per kilogram of seawater. (Source: Hartman, $D$, Global Physical Climatology, Academic Press.)
Estimate the salinity at a depth of 500 meters.

Inverse, Exponential, and Logarithmic Functions

Exponential and Logarithmic Equations and Inequalities

00:23

A Graphical Approach to College Algebra

Use any method (analytic or graphical) to solve each equation.
$$\ln \left(\ln e^{-x}\right)=\ln 3$$

Inverse, Exponential, and Logarithmic Functions

Exponential and Logarithmic Equations and Inequalities

01:48

A Graphical Approach to College Algebra

For the given $f(x)$, solve the equation $f(x)=0$ analytically and then use a graph of $y=f(x)$ to solve the inequalities $f(x)<0$ and $f(x) \geq 0$
$$f(x)=7-5 \log x$$

Inverse, Exponential, and Logarithmic Functions

Exponential and Logarithmic Equations and Inequalities

Applications and Modeling with Exponential and Logarithmic Functions

8 Practice Problems

04:19

A Graphical Approach to College Algebra

In real life, populations of bacteria, insects, and animals do not continue to grow indefinitely. Initially, population growth may be slow. Then, as their numbers increase, so does the rate of growth. After a region has become heavily populated or saturated, the population usually levels off because of limited resources. This type of growth may be modeled by a logistic function represented by
$$f(x)=\frac{c}{1+a e^{-b x}}$$
where $a, b,$ and $c$ are positive constants.

As age increases, so does the likelihood of coronary heart disease (CHD). The fraction of people $x$ years old with some CHD is approximated by
$$f(x)=\frac{0.9}{1+271 e^{-0.122 x}}$$
(Source: Hosmer, D. and S. Lemeshow, Applied Logistic Regression, John Wiley and Sons.)
(a) Evaluate $f(25)$ and $f(65) .$ Interpret the results.
(b) At what age does this likelihood equal $50 \% ?$

Inverse, Exponential, and Logarithmic Functions

Further Applications and Modeling with Exponential and Logarithmic Functions

02:26

A Graphical Approach to College Algebra

Suppose that the concentration of a bacteria sample is $100,000$ bacteria per milliliter. If the concentration doubles every 2 hours, how long will it take for the concentration to reach $350,000$ bacteria per milliliter?

Inverse, Exponential, and Logarithmic Functions

Further Applications and Modeling with Exponential and Logarithmic Functions

01:17

A Graphical Approach to College Algebra

Use the formula
$$y=R\left[\frac{1-(1+i)^{-(n-x)}}{i}\right]$$
where $y$ is the unpaid balance after $x$ payments have been made on a loan with $n$ payments of $R$ dollars. Find the balance after 120 payments have been made on the loan in Exercise 41 .

Inverse, Exponential, and Logarithmic Functions

Further Applications and Modeling with Exponential and Logarithmic Functions