Exponential and Logarithmic Functions Chapter 3 Exponential and Logarithmic Functions Copyright © 2014, 2010 Pearson Education, Inc.

Section 3.1 Exponential Functions Define an exponential function. Graph exponential functions. Develop formulas for simple and compound interest. Understand the number e. Graph natural exponential function. Model with exponential functions.

EXPONENTIAL FUNCTION A function f of the form is called an exponential function with base a and exponent x. Its domain is (–∞, ∞).

Example: Evaluating Exponential Functions d. Let F(x) = 4x. Find F(3.2).

Example: Evaluating Exponential Functions d. F(3.2) = 43.2 ≈ 84.44850629

RULES OF EXPONENTS Let a, b, x, and y be real numbers with a > 0 and b > 0. Then

Example: Graphing an Exponential Function with Base a > 1 Graph the exponential function Make a table of values. Plot the points and draw a smooth curve.

Example: Graphing an Exponential Function with Base a > 1 This graph is typical for exponential functions when a > 1.

Example: Graphing an Exponential Function f(x) = ax, with 0 < a < 1 Sketch the graph of Make a table of values. Plot the points and draw a smooth curve.

Example: Graphing an Exponential Function f(x) = ax, with 0 < a < 1 As x increases in the positive direction, y decreases towards 0.

PROPERTIES OF EXPONENTIAL FUNCTIONS Let f (x) = ax, a > 0, a ≠ 1. 1. The domain of f (x) = ax is (–∞, ∞). 2. The range of f (x) = ax is (0, ∞); the entire graph lies above the x-axis.

3. Since the y-value change by a factor of a for each unit increase in x. 4. For a > 1, the growth factor is a. Because the y-values increase by a factor of a for each unit increase in x. (i) f is an increasing function, so the graph rises to the right. (ii) as x → ∞, y → ∞. (iii) as x→ – ∞, y → 0.

5. For 0 < a < 1, the decay factor is a 5. For 0 < a < 1, the decay factor is a. Because the y-values decrease by a factor of a for each unit increase in x. (i) f is a decreasing function, so the graph falls to the right. (ii) as x → – ∞, y → ∞. (iii) as x → ∞, y → 0. 6. Each exponential function f is one-to-one. So, (i) if then m = n. (ii) f has an inverse.

7. The graph of f (x) = ax has no x-intercepts, so it never crosses the x-axis. No value of x will cause f (x) = ax to equal 0. 8. The graph of f (x) is a smooth and continuous curve, and it passes through the points 9. From 4(iii) and 5(iii), we conclude that the x-axis is a horizontal asymptote for every exponential function of the form f(x) = ax. 10. The graph of y = a-x is the reflection about the y-axis of the graph of y = ax.

Definitions The interest rate is the percent charged for the use of the principal for the given period. The interest rate, denoted by r, is expressed as a decimal. Unless stated otherwise, the period is assumed to be one year; that is, r is an annual rate. The amount of interest computed only on the principal is called simple interest.

SIMPLE INTEREST FORMULA The simple interest I on a principal P at a rate r (expressed as a decimal) per year for t years is

Example: Calculating Simple Interest Juanita has deposited $8000 in a bank for five years at a simple interest rate of 6%. How much interest will she receive? How much money will be in her account at the end of five years? a. Use the simple interest formula with P = $8000, r = 0.06, and t = 5.

Example: Calculating Simple Interest b. In five years, the amount A she will receive is the original principal plus the interest earned:

Definitions Compound interest is the interest paid on both the principal and the accrued (previously earned) interest. Interest that is compounded annually is paid once a year. For interest compounded annually, the amount A in the account after t years is given by

Example: Calculating Simple Interest Juanita deposits $8000 in a bank at the interest rate of 6% compounded annually for five years. How much money will she have in her account after five years? b. How much interest will she receive?

Example: Calculating Simple Interest a. Here P = $8000, r = 0.06, and t = 5. b. Interest = A  P = $10,705.80  $8000 = $2705.80.

COMPOUND INTEREST FORMULA A = amount after t years P = principal r = annual interest rate (expressed as a decimal) n = number of times interest is compounded each year t = number of years

Example: Using Different Compounding Periods to Compare Future Values If $100 is deposited in a bank that pays 5% annual interest, find the future value A after one year if the interest is compounded annually. semiannually. quarterly. monthly. daily.

Example: Using Different Compounding Periods to Compare Future Values In the following computations, P = 100, r = 0.05 and t = 1. Only n, the number of times interest is compounded each year, changes. Since t = 1, nt = n(1) = n. (i) Annual Compounding:

Example: Using Different Compounding Periods to Compare Future Values (ii) Semiannual Compounding: (iii) Quarterly Compounding:

Example: Using Different Compounding Periods to Compare Future Values (iv) Monthly Compounding: (v) Daily Compounding:

CONTINUOUS COMPOUND INTEREST FORMULA A = amount after t years P = principal r = annual interest rate (expressed as a decimal) t = number of years

Example: Calculating Continuous Compound Interest Find the amount when a principal of $8300 is invested at a 7.5% annual rate of interest compounded continuously for eight years and three months. P = $8300 and r = 0.075. Convert eight years and three months to 8.25 years.

THE NATURAL EXPONENTIAL FUNCTION The exponential function with base e is so prevalent in the sciences that it is often referred to as the exponential function or the natural exponential function.

THE NATURAL EXPONENTIAL FUNCTION

Example: Transformations on f(x) = ex Use transformations to sketch the graph of Shift the graph of f (x) = ex one unit right and two units up.

MODEL FOR EXPONENTIAL GROWTH OR DECAY A(t) = amount at time t A0 = A(0), the initial amount k = relative rate of growth (k > 0) or decay (k < 0) t = time

Example: Exponential Growth In the year 2000, the human population of the world was approximately 6 billion and the annual rate of growth was about 2.1%. Using the model on the previous slide, estimate the population of the world in the following years. 2030 1990

Example: Exponential Growth a. The year 2000 corresponds to t = 0. So A0 = 6 (billion), k = 0.021, and 2030 corresponds to t = 30. The model predicts that if the rate of growth is 2.1% per year, over 11.26 billion people will be in the world in 2030.

Example: Exponential Growth b. The year 1990 corresponds to t = 10. The model predicts that the world had over 4.86 billion people in 1990. (The actual population in 1990 was 5.28 billion.)

Section 3.2 Logarithmic Functions Define logarithmic functions. Evaluate logarithms. Find the domains of logarithmic functions. Graph logarithmic functions. Use logarithms to evaluate exponential equations.

DEFINITION OF THE LOGARITHMIC FUNCTION For x > 0, a > 0, and a ≠ 1, The function f (x) = loga x, is called the logarithmic function with base a. The logarithmic function is the inverse function of the exponential function.

Example: Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form.

Example: Converting from Logarithmic Form to Exponential Form Write each logarithmic equation in exponential form.

Example: Evaluating Logarithms Find the value of each of the following logarithms.

Example: Evaluating Logarithms

Example: Using the Definition of Logarithms Solve each equation.

Example: Using the Definition of Logarithms

Example: Using the Definition of Logarithms

DOMAIN OF LOGARITHMIC FUNCTION Domain of y = loga x is (0, ∞) Range of y = loga x is (–∞, ∞) Logarithms of 0 and negative numbers are not defined.

Example: Finding Domains Find the domain of Domain of a logarithmic function must be positive, that is, The domain of f is (–∞, 2).

BASIC PROPERTIES OF LOGARITHMS For any base a > 0, with a ≠ 1, loga a = 1. loga 1 = 0. loga ax = x, for any real number x.

Example: Sketching a Graph Sketch the graph of y = log3 x. Plotting points (Method 1)

Example: Sketching a Graph Plot the ordered pairs and connect with a smooth curve to obtain the graph of y = log3 x.

Example: Sketching a Graph Inverse function (Method 2) Graph y = f (x) = 3x. Reflect the graph of y = 3x in the line y = x to obtain the graph of y = f –1(x) = log3 x.

PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS Exponential Function f (x) = ax Logarithmic Function f (x) = loga x Domain (–∞, ∞) Range (0, ∞) Domain (0, ∞) Range (–∞, ∞) y-intercept is 1 No x-intercept x-intercept is 1 No y-intercept x-axis (y = 0) is the horizontal asymptote y-axis (x = 0) is the vertical asymptote

PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS Exponential Function f (x) = ax Logarithmic Function f (x) = loga x Function is one-to-one. au = av if and only if u = v. The function is one-to-one. logau = logav if and only if u = v. The function is increasing if a > 1 and decreasing of 0 < a < 1. The function is increasing if a > 1 and decreasing of 0 < a < 1.

GRAPHS OF LOGARITHMIC FUNCTIONS

Example: Using Transformations Start with the graph of f (x) = log3 x and use transformations to sketch the graph of each function. State the domain and range and the vertical asymptote for the graph of each function.

Example: Using Transformations Shift up 2 Domain (0, ∞) Range (–∞, ∞) Vertical asymptote x = 0

Example: Using Transformations Shift right 1 Domain (1, ∞) Range (–∞, ∞) Vertical asymptote x = 1

Example: Using Transformations Reflect graph of y = log3 x in the x-axis Domain (0, ∞) Range (–∞, ∞) Vertical asymptote x = 0

Example: Using Transformations Reflect graph of y = log3 x in the y-axis Domain (∞, 0) Range (–∞, ∞) Vertical asymptote x = 0

COMMON LOGARITHMS log 10 = 1 log 1 = 0 log 10x = x The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: log x = log10 x. Thus, y = log x if and only if x = 10 y. Applying the basic properties of logarithms log 10 = 1 log 1 = 0 log 10x = x

Example: Using Transformations to Sketch a Graph Sketch the graph of Start with the graph of f (x) = log x. Step 1: Replacing x with x – 2 shifts the graph two units right.

Example: Using Transformations to Sketch a Graph Step 2: Multiplying by 1 reflects the graph in Step 3: Adding 2 shifts the graph two units up. the x-axis.

NATURAL LOGARITHMS ln e = 1 ln 1 = 0 log ex = x The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = loge x. Thus, y = ln x if and only if x = e y. Applying the basic properties of logarithms ln e = 1 ln 1 = 0 log ex = x

Example: Evaluating the Natural Logarithm Function Evaluate each expression. (Use a calculator.)

Example: Doubling Your Money How long will it take to double your money if it earns 6.5% compounded continuously? b. At what rate of return, compounded continuously, would your money double in 5 years? If P is the original amount invested, A = 2P. It will take 11 years to double your money.

Example: Doubling Your Money Your investment will double in 5 years at the rate of 13.86%.

NEWTON’S LAW OF COOLING Newton’s Law of Cooling states that where T is the temperature of the object at time t, Ts is the surrounding temperature, and T0 is the value of T at t = 0.

Example: McDonald’s Hot Coffee The local McDonald’s franchise has discovered that when coffee is poured from a coffeemaker whose contents are 180ºF into a noninsulated pot, after 1 minute, the coffee cools to 165ºF if the room temperature is 72ºF. How long should the employees wait before pouring the coffee from this noninsulated pot into cups to deliver it to customers at 125ºF?

Example: McDonald’s Hot Coffee Use Newton’s Law of Cooling with T0 = 180 and Ts = 72 to obtain We have T = 165 and t = 1.

Example: McDonald’s Hot Coffee Substitute this value for k. Solve for t when T = 125. The employee should wait about 5 minutes.

GROWTH AND DECAY MODEL A is the quantity after time t. A0 is the initial (original) quantity (when t = 0). r is the growth or decay rate per period. t is the time elapsed from t = 0.

Example: Chemical Toxins in a Lake In a large lake, one-fifth of the water is replaced by clean water each year. A chemical spill deposits 60,000 cubic meters of soluble toxic waste into the lake. How much of this toxin will be left in the lake after four years? b. When will the toxic chemical be reduced to 6000 cubic meters?

Example: Chemical Toxins in a Lake One-fifth of the water in the lake is replaced by clean water every year, so the decay rate, r is and A0 = 60,000. So, where A is the amount of toxin (in cubic meters) after t years.

Example: Chemical Toxins in a Lake a. Substitute t = 4. b. Substitute A = 6000 and solve for t.

Section 3.3 Rules of Logarithms Learn the rules of logarithms. Estimate a large number. Change the base of a logarithm. Apply logarithms in growth and decay.

RULES OF LOGARITHMS Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. 1. Product Rule: The logarithm of the product of two (or more) numbers is the sum of the logarithms of the numbers.

RULES OF LOGARITHMS Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. 2. Quotient Rule The logarithm of the quotient of two (or more) numbers is the difference of the logarithms of the numbers.

RULES OF LOGARITHMS Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. 3. Power Rule The logarithm of a number to the power r is r times the logarithm of the number.

Example: Using Rules of Logarithms to Evaluate Expressions Given that log 5 z = 3 and log 5 y = 2, evaluate each expression.

Example: Using Rules of Logarithms to Evaluate Expressions

Example: Using Rules of Logarithms to Evaluate Expressions

Example: Writing Expressions in Expanded Form Write each expression in expanded form.

Example: Writing Expressions in Expanded Form

Example: Writing Expressions in Condensed Form Write each expression in condensed form.

Example: Writing Expressions in Condensed Form

Example: Writing Expressions in Condensed Form

Example: Writing Expressions in Condensed Form

CHANGE-OF-BASE FORMULA Let a, b, and x be positive real numbers with a ≠ 1 and b ≠ 1. Then logb x can be converted to a different base as follows:

Example: Using a Change of Base to Compute Logarithms Compute log513 by changing to a. common logarithms and b. natural logarithms.

HALF-LIFE FORMULA The half-life of any quantity whose value decreases with time is the time required for the quantity to decay to half its initial value. The half-life of a substance undergoing exponential decay at a rate k (k < 0) is given by the formula 89

Example: Finding the Half-Life of a Substance In an experiment, 18 grams of the radioactive element sodium-24 decayed to 6 grams in 24 hours. Find its half-life to the nearest hour. So 90

Example: Finding the Half-Life of a Substance Use the formula 91

Section 3.4 Exponential and Logarithmic Equations and Inequalities Solve exponential equations. Solve applied problems involving exponential equations. Solve logarithmic equations. Use the logistic growth model. Use logarithmic and exponential inequalities

Example: Solving an Exponential Equation Solve each equation. 93

Example: Solving Exponential Equations Using Logarithms Solve exponential equations when both sides are not expressed with the same base. Step 1 Isolate the exponential expression on one side of the equation. Step 2 Take the common or natural logarithm of both sides. Solve for x: 5 ∙ 2x – 3 = 17. 1. 2.

Example: Solving Exponential Equations Using Logarithms Step 3 Use the power rule, logaMr = r loga M. Step 4 Solve for the variable. 3. 4.

Example: Solving an Exponential Equation with Different Bases Solve the equation 52x–3 = 3x+1 and approximate the answer to three decimal places. When different bases are involved, begin with Step 2. 96 96

Example: An Exponential Equation of Quadratic Form Solve the equation 3x – 8 ∙ 3–x = 2. This equation is quadratic in form. Let y = 3x; then y2 = (3x)2 = 32x.

Example: An Exponential Equation of Quadratic Form But 3x = –2 is not possible because 3x > 0 for all numbers x. So, solve 3x = 4.

Example: An Exponential Equation of Quadratic Form

Annual Population Growth Rate Example: Solving a Population Growth Problem The following table shows the approximate population and annual growth rate of the United States and Pakistan in 2010. Country Population Annual Population Growth Rate United States 308 million 0.9% Pakistan 185 million 2.1%

Example: Solving a Population Growth Problem Use the alternate population model P = P0(1 + r)t, where P0 is the initial population and t is the time in years since 2010. Assume that the growth rate for each country stays the same. Estimate the population of each country in 2020. In what year will the population of the United States be 350 million? In what year will the population of Pakistan be the same as the population of the United States?

Example: Solving a Population Growth Problem Use the given model a. The U.S. population in 2010 was P0 = 308. In ten years, 2020, the population of the U.S. will be Pakistan population in 2020 will be

Example: Solving a Population Growth Problem b. Solve for t to find when the United States population will be 350. The population of the U.S. will be 350 million in approximately 14.27 years after 2010, sometime in 2024.

Example: Solving a Population Growth Problem c. Solve for t to find when the population will be the same in the two countries.

Example: Solving a Population Growth Problem The two populations will be equal in about 43.12 years, that is, during 2053.

SOLVING LOGARITHMS EQUATIONS Equations that contain terms of the form log a x are called logarithmic equations. To solve a logarithmic equation we write it in the equivalent exponential form.

Example: Solving a Logarithmic Equation Solve: We must check our solution.

Example: Solving a Logarithmic Equation ? ? Check x = The solution set is

Example: Using the One-to-One Property of Logarithms Solve

Example: Using the One-to-One Property of Logarithms Check x = 2 ? ? ?

Example: Using the One-to-One Property of Logarithms Check x = 3 ? ? ? The solution set is {2, 3}.

Example: Using the Product and Quotient Rules Solve: 112

Example: Using the Product and Quotient Rules Check x = 2 ? ? Logarithms are not defined for negative numbers, so x = 2 is not a solution. 113

Example: Using the Product and Quotient Rules Check x = 5 ? ? ? The solution set is {5}. 114

Example: Using the Product and Quotient Rules b. 115

Example: Using the Product and Quotient Rules Check x = 2 Check x = 5 log2 (1) and log2 (2) are undefined, so solution set is { –2}. 116

Example: Using Logarithms in the Logistic Growth Model Suppose the carrying capacity M of the human population on Earth is 35 billion. In 1987, the world population was about 5 billion. Use the logistic growth model of P. F. Verhulst to calculate the average rate, k, of growth of the population, given that the population was about 6 billion in 2003. 117

Example: Using Logarithms in the Logistic Growth Model We have t = 0 (1987), P(t) = 5 and M = 35. We now have 118

Example: Using Logarithms in the Logistic Growth Model Solve for k given t = 16 (for 2003) and P(t) = 6. The growth rate was approximately 1.35%. 119

Example: Solving an Inequality Involving an Exponential Expression Solve: 5(0.7)x + 3 < 18 Reverse the sense of the inequality since ln(0.7) < 0. 120

Example: Solving an Inequality Involving a Logarithmic Expression Solve: log (2x – 5) ≤ 1 Find the domain for log (2x – 5). Because 2x – 5 must be positive, solve 2x – 5 > 0; so 2x > 5 or 121

Example: Solving an Inequality Involving a Logarithmic Expression Then Combining with we find that the solution set is 122

Section 3.5 Logarithmic Scales Define pH. Define Richter scale for measuring earthquake intensity. Define scales for measuring sound.

pH SCALE FORMULA We define the pH scale of a solution by the formula Where [H+] is the concentration of H+ ions in moles per liter. (A mole is a unit of measurement, equal to 6.023 × 1023 atoms.)

Example: Calculating pH and [H+] a. Calculate to the nearest tenth of pH value of grapefruit juice if [H+] of grapefruit juice is 6.23 × 10–4.

Example: Calculating pH and [H+] b. Find the hydrogen concentration [H+] in beer if its pH value is 4.82.

Example: Acid Rain How much more acidic is acid rain with a pH value of 3 than an ordinary rain with a pH value of 6? We have so that Similarly, Therefore. The hydrogen ion concentration is this acid rain is 1000 times greater than in ordinary rain.

THE RICHTER SCALE The magnitude M of an earthquake is a function of its intensity I and is defined by where I0 is a zero-level earthquake 128

Example: Comparing Two Earthquakes Compare the intensity of the Mexico City earthquake of 1985 which registered 8.1 on the Richter scale to that of the San Francisco earthquake of 1989 which measured 6.9 on the Richter scale. Let IM and IS denote the intensities of the Mexico City and San Francisco earthquakes, respectively. 129

Example: Comparing Two Earthquakes Divide IM by Is and simplify. The intensity of the Mexico City earthquake was about 16 times that of the San Francisco earthquake. 130

ENERGY OF EARTHQUAKES The energy E (in joules) released by an earthquake of magnitude M (Richter scale) is given by or

Example: Comparing Two Earthquakes Compare the estimated energies released by the San Francisco earthquake of 1906 and the Northridge earthquake of 1994. Using the table on the previous slide M1906 = 7.8 and M1994 = 6.7, and the earthquakes we have The 1906 earthquake released more than 45 times as much energy as the 1994 earthquake.

LOUDNESS OF SOUND AND DECIBELS The loudness (or relative intensity) L of a sound measured in decibels is related to its intensity I by the formula or where

Example: Comparing Intensities How much more intense if a 65 dB sound than a 42 dB sound? Thus, a sound of 65 dB is about 200 times more intense than a sound of 42 dB.

Example: Computing Sound Intensity Calculate the intensity on watts per square meter of a sound of 73 dB.