Exponential and logarithmic word problems worksheet pdf. 1. Use your model to estimate the population for this year. It contains 28 problems involving derivatives of functions including ln(x), ex, log(x), and combinations of exponentials, logarithms, and polynomial terms. 2. Example: Nov 16, 2022 · Here is a set of practice problems to accompany the Logarithm Functions section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University. Each problem is followed by specific questions and their corresponding solutions. 1 Exponential and Logarithmic Functions Practice Problems 1. Please note that the final answer can be represented in numerous ways. (b) Do the outputs show the population is tripling every 24 hours (1 day?) Logarithm can be used to solve word problems. Word Problems: Interest, Growth/Decay, and Half-Life Applying logarithms and exponential functions Topics include simple and compound interest, e, depreciation, rule of 72, exponential vs. You should also try to understand how changing any one of these a ects the 10. This means half of a sample of C 14 will decay every 5700 years. P S AAPlolZ XrMiKgNhQtAsp ar8eusSecrlvnevdT. convert between exponential and logarithmic forms. Simplify the following expression: 3. So, the correct way to solve these types of logarithmic problems is to rewrite the logarithmic problem in exponential form. On the other hand, exponential word problems tend to be much more involved, requiring, among other things, that the student first Algebra 2 Log Word problems Name: Date: Logarithmic Word Problems Complete the following questions on separate paper. **Check for extra ous solutions before moving on from each problem. EXPONENTIAL & LOGARITHMIC EQUATIONS Solve each equation. Then use the value of x to rewrite the exponential equation in its equivalent logarithmic form, x logb y. 4. Create your own worksheets like this one with Infinite Precalculus. Exponential and Logarithmic Applications Worksheet 1. My view is that if you have a Log and Exponential Word Problems (Day 2) - Tuesday, 4/21 For each: Write the formula, show substitution into the formula, solve until in calculator ready form. In the same year, another earthquake was recorded that measured six time stronger on the Richter scale. Math 221 Week 5 part 3 Word problems with exponentials Please take a moment to just breathe. Pre-Calculus Math 30-1 Exponential and Logarithmic Word Problems Solve the following problems algebraically, show all work including setting up the equation for the problem. 4: Exponential Equations, Exponential Growth and Decay, Express Exponentials as Logarithms 7. Create your own worksheets like this one with Infinite Algebra 2. Assistance in exponential and logarithmic word worksheet, and equation is a purchase by offering discount codes on the same number of denominators, the content and the worksheets. You may select the number of problems, the types of problems, and the variable to use. . Exponential & Logarithm worksheets. If we consider the example log (7x What You Will Learn Solve exponential equations with the same base. Prove the following statements. Nov 16, 2022 · Here is a set of practice problems to accompany the Solving Logarithm Equations section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University. 86) where A is the amount after depreciation. This hands-on poster project reinforces key math concepts while encouraging creativity, collaboration, and critical They apply the Change of Base Formula to rewrite a logarithm using a different base, and they apply appropriate formulas to solve problems involving exponential growth and exponential decay. In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. If so, the Section 7. Printable in convenient PDF format. This involves very large or very small values. 5 This Algebra 2 Exponential Equations Not Requiring Logarithms worksheet will give you exponential equations to solve. 6. Home > Math Topics > Logarithms > Logarithm Word Problems Worksheets The primary power of logarithms lies in their ability to help use model and understand exponential situations. Solve exponential growth and decay problems. com It includes problems involving evaluating logarithmic expressions, writing expressions in logarithmic and exponential form, solving logarithmic equations, graphing logarithmic functions, finding inverses of functions, and applying logarithms to word problems involving compound interest and exponential growth. How many flies will be there after days? Create your own worksheets like this one with Infinite Algebra 2. LE. Graph and translate a logarithmic function. Some of the worksheets displayed are Work logarithmic function, Mixed logarithms word problems, Chapter 12 logarithms, Work 2 7 logarithms and exponentials, Solving exponential and logarithmic equations, Solving logarithmic equations, Meaning of logarithms, Properties of The exponential and logarithmic word problems worksheet pdf includes a variety of problems designed to improve your understanding of these concepts. Logarithms with Base a ∗ Definition ∗ Exponential Notation vs. Exponential Equations and Logarithms Solve each equation for the unknown variable. The amount of money in her bank account grows at 2. Showing top 8 worksheets in the category - Logarithm Word Problems. Name: Date: Unit 7: Exponential & Logarithmic Functions Homework 5: Graphing Logarithmic Functions ** This is a 2-page document! ** Directions: Graph each function and identify its key characteristics. True or False? 7. 7 from module 2. Apr 4, 2015 · Practice Worksheet: Evaluating Logarithms Rewrite the equation in exponential form. If the annual growth rate averaged about 1. You can choose the number of problems you want and the form of the problems. The questions in these worksheets include direct application of formulas, conversion of exponents to logarithms, and vice versa as well as work problems. Includes anticipation guide, graph analysis, and growth/decay problems. Word Problems - Logarithms 1) The amount g (in trilliions of cubic feet) of natural gas consumed in the United States from 1980 to 2010 can be modeled by: g = 2. 3% per year, write an exponential equation that models this situation. Free trial available at KutaSoftware. Write the following expressions in terms of logs of x, y and z. 326 Solving Exponential Equations with the Same Base Exponential equations are equations in which variable expressions occur as exponents. Create your own worksheets like this one with Infinite Algebra 1. The value after t years is given by ( ) = 60 000(0. Round your answers to the nearest ten-thousandth. How long, to the nearest year, will it take for the value of the car to depreciate to $18 000? 2. There are 2 graphs, 2 tables, 2 function rules, and 2 real world situations for each type. x g eAPljlx prVidgchjtus6 gr5eBsmeQrtvQetdo. 001 k log16 2 l ©4 v210y1T3A cK3uTtQaL OSEof 8tbwGa2rweY VL6LnCD. Jun 6, 2018 · Here are a set of practice problems for the Exponential and Logarithm Functions chapter of the Algebra notes. Exponential Function Word Problems Exponential growth is modelled by y = y0ekt There are four variables, the initial amount, y0, the time t, the growth factor k, and the current amount y: You should be comfortable with nding any one of these four, given the other three. A logarithmic scale uses powers of 10 to compare numbers that vary greatly in size. When solving application problems that involve exponential and logarithmic functions, we need to pay close attention to the position of the variable in the equation to determine the proper way solve the equation we investigate solving equations that contain exponents. Make exponential functions engaging and interactive with this real-world word problem activity! Students will explore exponential growth and decay through fun, relatable scenarios involving zombies, cockroaches, and bacteria. In this section: We investigate the differential equation y′ = . Find a. Students are asked to write logarithmic and exponential equations in alternate forms, simplify expressions, solve equations for variables in terms of logarithms and exponents, and perform other What Are Logarithms? As you move ahead in the field of advanced mathematics, you will come across problems that you must solve quickly to move ahead. Advanced Functions: Chapter 7: Practice Problems - Exponential and Logarithmic Functions Here are some practice problems! Some of these are easy, some aren’t. Solve exponential equations by graphing. She can expect a 6% raise in pay each year. A logarithm is defined as the power of which a number needs to be raised to get another number. Exponential and Logarithmic Functions Recall from worksheet 3. With our equation of: log 4(q) = m Worksheet – Applications of Exponential and Logarithmic Functions 1. After days there are flies, and after days there are flies. If a population of 2350 grows to 7000 in 32 years, what is the value of b? Exponential, Logarithmic and other Word Problems HW Date________________ Period____ Find the inverse of each function. 3% per year. The student will be given functions and will be asked to find their indefinite integral. a) Write an exponential equation to represent the population as a function of time. Suppose a culture of bacteria is put in a petri dish and grows continuously. Write the exponential regression equation, expressing the regression coefficients to the nearest tenth. These worksheets help students practice solving equations that involve logarithms, enhancing their understanding of logarithmic functions and applications. There are 26 problems in total asking to solve equations, find original investment amounts, and calculate time for an account to reach a certain value with compound interest. Write the equation in terms of x, the number of years since 1963. A sports car priced at $60 000 depreciates at a rate of 14% per year. 5 7 RM0aOdaeB Tw8iCtOhe TI0njfDiznuiHtzee FAdl2g9eTbAraaQ W2k. These Logarithmic and Exponential Integrals Worksheets are a great resource for Logarithm Worksheets Explore our logarithm worksheets with answer keys and make great progress in rewriting expressions in exponential and logarithmic forms. Malcolm bought a new car for $24 000. *The graphs, tables, rules, and story problems are not related. If a sample originally contained 40 grams of C 14, find an expression for N, the amount of C 14 remaining after t years. Expand (x + y)5. Free Algebra 2 worksheets created with Infinite Algebra 2. Evaluate the following powers: 130 = , ( 8)2/3 = − Exponential and Logarithmic Word problems 1 A microbiologist is studying two bacteria populations. My mother gave me her college algebra book. 02) t , where t represents the number of years since the current population count. 13A Logarithmic and Exponential Equations and Inequalties Part A - YouTube Jan 27, 2025 · Looking for practice materials to master exponential and logarithmic equations? Our comprehensive worksheets provide ample opportunities to hone your skills! Each worksheet is designed with varied problems, accompanied by a detailed answer key in PDF format for easy checking. Evaluate logarithmic expressions with and without a calculator. Rewrite each equation in logarithmic form. I learned from it how to solve word problems, although I remember distinctly that I never really understood them. 5 Practice - Logarithmic Functions Rewrite each equation in exponential form. For ln 100 ln 4 example, log5 100 can be re-written as . You should be able to. Logarithmic Functions - You will be asked to convert values to exponential form, determine ranges, and condense values. Rewrite each equation in exponential form. Create your own worksheets like this one with Infinite Calculus. and Mrs. 5. worksheets with answers. 8. Solving Logarithmic Equations (Word Problems) Example 1 INVESTMENT Mr. 2) The second problem involves sound intensity measured in decibels. com I developed a proficiency [in junior high school] in simple algebra that lasted for a long time and has been very useful. The graph of an exponential or logarithmic function can be used to predict the greatest and least instantaneous rates of change and when they occur. 1) The first problem determines the magnitude on the Richter scale of an earthquake 1,000 times more intense than the standard intensity. The population of a small town can be modelled by the exponential function P(t) = 5000(1. exponential equation, p. Suppose we have an equation in the Rewriting Exponential Equations Work with a partner. Logarithm worksheets for high school students cover the skills based on converting between logarithmic form and exponential form, evaluating logarithmic expressions, finding the value of the variable to make the equation correct, solving logarithmic equations, single logarithm, expanding logarithm using power rule, product rule and quotient rule, expressing the log value in algebraic Logarithmic Equations - You can employ some algebra skills here to find unknowns. com 5. Tanya currently has $4000 in her bank account. Solve each exponential equation. 10 that the derivative of ex is ex. Logarithmic Notation ∗ Evaluating Logarithms ∗ Graphs of Logarithms ∗ Domain of Logarithms Properties of Logarithms ∗ Simplifying Logarithmic Expressions ∗ Using the Change of Base Formula to Find Approximate Values of Logarithms Solving Exponential and Logarithmic Browse word problems exponential and logs resources on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources. com These worksheets demonstrate the steps required to solve exponential equations and give practice problems to help students master the skill. linear models, and more. The town historian records the population at the end of each year. Write the following equalities in exponential form. Master the rules demonstrated by log equations to expand and condense, while also honing children’s skills in evaluating common and natural logarithms. Word problems involving surds. 3, is all about Logarithmic Functions. If this decrease continued, during what year will the population reach 4,000 people? Logarithmic word problems often involve finding pH, sound levels in decibels, and the force of an earthquake according to the Richter scale. Using the power of a power property of exponential functions, we can multiply the exponents: These worksheets are structured to guide students through how to solve logarithmic equations and convert between exponential or logarithmic forms. To solve a logarithmic equation, first isolate the logarithmic expression, then We would like to show you a description here but the site won’t allow us. com Free Precalculus worksheets created with Infinite Precalculus. For example, the rule = 4 − 8 will not have a matching table – you must decide if this rule is linear, exponential growth, or exponential The document contains word problems involving logarithmic functions and exponential growth/decay. so, the argument of a logarithm must be positive. These Calculus Worksheets will produce problems that involve using the logarithmic and exponential rules to find indefinite integrals. 1. This document contains a natural log worksheet with 38 problems involving evaluating, simplifying, solving equations, and other operations with natural logarithms and exponential functions. ©S C2b0U1725 TKruAtGah lSKoofltIwfaSr6eC OLzLtCD. If you’d like a pdf document containing the solutions the download tab above contains links to pdf’s containing the solutions for the full book, chapter and section. Exponential and logarithmic equations word problems worksheet When addressing the application problems associated with exponential and logarmic functions, we need to pay close attention to the position of the variable in the equation to determine the correct way to solve the equation that we are exploring by solving the equations that contain the exhibitors. each year, write and exponential function to model the situation, then find the value of the care in 2020. One of those problems is the logarithm. Upon graduating from college, Ashley took a job with a starting salary of $40,000. Mastering Logarithmic & Exponential Equations - Free Practice WorksheetProduct Description:Boost your students' confidence in solving logarithmic and exponential equations with this FREE practice worksheet! This worksheet includes 10 thoughtfully crafted questions across two key topics:Solving E 1. Algebra 2/Pre-Calculus Exponential Word Problems (Day 2, Exponentials) Name__________________ In this handout, we will continue our study of exponential word problems. In this worksheet we will consider the special relationship logarithms and exponentials have with each other. The magnitude is 3. Exponential and Logarithmic Functions Practice Exam - ANSWER KEY Video solutions are in italics. g N WA_l[lU Mr\iigBhztPs` RrleKsReGruvFeSdR. Nov 16, 2022 · Here is a set of practice problems to accompany the Solving Exponential Equations section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University. 3. You can chose the form of the problems. Pre-Calculus Math 30-1 Exponential and Logarithmic Word Problems Solve the following problems algebraically, show all work including setting up the equation for the problem. Mitchell are saving for their daughter’s college education. There are two main types of difference between exponential equations problems, and it entirely depends on the bases. I could do only the problems that followed the pattern of the examples in the book. In 1‐9, solve each exponential equation. Logarithm Word Problems Worksheets How to Tackle Logarithms in Word Problems Logarithmic word problems, in my experience, generally involve evaluating a given logarithmic equation at a given point and solving for a given variable; they're pretty straightforward. 2 and an antacid tablet has a pH of 10. Core Vocabulary Solve exponential equations with unlike bases. The If we consider the problem log (5x 2 + 7) = 5, this problem contains a term, 5, that does not have a logarithm. We solve word problems using the exponential and logarithmic functions. A: Concepts Exercise \ (\PageIndex {1}\) 1) How can an exponential equation be solved? 2) When does an extraneous solution occur? How can an extraneous solution be recognized? 3) When can the one-to-one property of logarithms be used to solve an equation? When can it not be used? Answer 1 Determine first if the equation can be rewritten so that each side uses the same base. JMAP F. Suppose we have an equation in the form: value = coefficient (base) exponent Section 3 The Natural Logarithm and Exponential The natural logarithm is often written as ln which you may have noticed on your calculator. To solve an exponential equation, first isolate the exponential expression, then take the logarithm of both sides of the equation and solve for the variable. Logarithmic Notation ∗ Evaluating Logarithms ∗ Graphs of Logarithms ∗ Domain of Logarithms Properties of Logarithms ∗ Simplifying Logarithmic Expressions ∗ Using the Change of Base Formula to Find Approximate Values of Logarithms Solving Exponential and Logarithmic Free Algebra 2 worksheets created with Infinite Algebra 2. For example, very small and very large concentrations of the hydrogen ion in a solution influence its classification as either a base or an acid ©y a2V0`1x5g JKJuitQa[ CSkoNfitDwTaErpeV gL]LSCH. It includes problems related to population growth, bouncing balls, radioactive decay, bank interest, depreciation of a laptop, and bacterial growth. Write an exponential function in the form y = abx that could be used to model the number of cars y in millions for 1963 to 1988. The graph of an exponential function g(x) = ax is shown below. We will learn that when we take the inverse of an exponential function we have a new type of function called a logarithmic function, and will draw comparisons between those functions and the ones they came from. Give an exact solution. examples and step by step solutions, Grade 7, mental math Solving Logarithm Equations Worksheet Solve each equation. Rewrite logarithmic expressions as exponential expressions (and vice-versa). This document contains math word problems involving exponential and logarithmic equations. This will leave us with an exponential equation. You'll find real-world applications, solutions, and examples to help clarify your learning. 91(1. ln x = loge x The symbol e symbolizes a special mathematical constant. On the accompanying grid, make a scatter plot of these data. The problems start with basic derivatives and progress to finding derivatives at specified points and of MadAsMaths :: Mathematics Resources Exponential and Logarithmic Functions Worksheets Exponential and logarithmic functions worksheets demonstrate how to solve problems based on exponents and logs using formulas and manipulations. It is also possible to write this number as 2 + log5 4 or 2 + ln 5 ln 5 because ln 4 log5 100 = log5 (25 · 4) = log5 25 + log5 4 = 2 + log5 4 = 2 + ln 5 ln 100 ln 4 All of these forms, log5 100 = = 2 + log5 4 = 2 + are acceptable as final Exponential function problems can be tricky, so I put together an exponential functions word problems worksheet with answers to help you out! CLASS EXAMPLES: Solve each equation. Meaning if the value you get for x makes the argument either 0 or a negative, you must exclude that value fro the solution Examples: Solve each Objectives: Students will be able to model word problems with exponential functions and use logs to solve exponential models. This is an extra source for revising the material for Exam 3. Ask for help, read your notes and your textbook, use your mind. Logarithmic Expressions - You will be asked to expand and evaluate expressions that are in logarithm form. The natural and physical science disciplines often use them to solve real world that relate to things like pitch of sound, intensity of light, and measures of Exponent and Logarithm Practice Problems for Precalculus and Calculus 1. Evaluate. The population of a town changes by an exponential growth factor, b, every 4 years. Last Monday, the 1st population numbered 2000 and the 2nd numbered 2 048 000. Function type of this exponential and logarithmic word problems worksheets will automatically after completing this project is. EARTHQUAKE PROBLEMS: Problem 1: Early in the century an earthquake measured 8. The rst problem here we are asked to rewrite the equation in exponential form. The formula we are going to use for exponential form is: Xa = b The meaning behind this formula is that x=base, a=exponent, and b=answer. Nov 16, 2022 · Here is a set of practice problems to accompany the Exponential and Logarithm Equations Section of the Review chapter of the notes for Paul Dawkins Calculus I course at Lamar University. 10. This new section, 4. Where necessary, give the exact value and then use a calculator to obtain a decimal approximation, correct to four decimal places. Convert the logarithmic equation to an exponential equation. = 109 x = logl x +3 = log4(x +5) 2. Solving Exponential and Logarithmic equations. Section 3 The Natural Logarithm and Exponential The natural logarithm is often written as ln which you may have noticed on your calculator. Introduction We have met exponentials and logarithms before in previous modules. He noted that the 1st population doubled every day while the 2nd population was reduced by half each day. We cover the common types of problems you run into with these worksheets. Evaluating Exponential Functions Worksheet PDF. 0 on the Richter scale. Rewriting Exponential Equations in Logarithmic Form The ability to convert between exponential and logarithmic forms is Final Exam Review Linear and Exponential Word Problems Writing Linear and Exponential Models: For the following four problems, take the information given and write rst a linear model and, second, write an exponential model with the given parameters. These Algebra 2 - Exponential and Logarithmic Functions Worksheets will give you equations to rewrite into either exponential or logarithmic form. Graphing simple rational functions Graphing general rational functions Simplifying rational expressions Multiplying / dividing rational expressions Adding / subtracting rational expressions Complex fractions Solving rational equations Exponential equations not requiring logarithms Exponential equations requiring logarithms Graphing logarithms This document contains a worksheet with multiple questions on logarithmic functions. After we isolate the logarithm we will then apply the definition. Topics include logarithm laws, graphing, exponential equations, growth and decay models, 1⁄2 life, and more The document is a practice worksheet for exponential and logarithmic equations with exercises and word problems for a 10th grade pre-calculus class taught by Estefanny Ruíz González. Y Aug 21, 2024 · So the solution is log5 100. 4B Word Problems Involving Logarithms Here will we look at application problems very similar to those back in Unit 6, except this time we will have to use logarithms to solve them. 1) 9log 9 v = 0 3) -7 - 10log r = -27 The following problems will help you in your study about Exponential and Logarithmic Functions and their Applications. Get, Create, Make and Sign exponential and logarithmic word problems worksheet pdf form Edit your logarithm word problems worksheet form online Type text, complete fillable fields, insert images, highlight or blackout data for discretion, add comments, and more. Here we will have to solve logarithmic equations so the process is a bit different. Note: Answers are given at the end of this problem. 9. EXAMPLE 1 Solving a problem using a logarithmic equation In chemistry, the pH (the measure of acidity or alkalinity of a substance) is based on a logarithmic scale. Solve the following logarithmic equations. 8: Curve Fitting with Exponential and Logarithmic Functions Learning Target: We are learning about creating equations to model exponential and logarithmic data. It then follows that the anti derivative of ex is ex: Linear versus Exponential card sort Sort them by Linear, Exponential Growth, and Exponential Decay. A certain fruit has a pH of 2. They invest $10,000 in an account that pays 4. 07)t where t equals the number of years since 1980. h M vMGapdneN 0wViOtqhC FIEntfLikn5i3tQeW cAJlCgYeDbcrnaD I2j. You may approximate answers using a calculator. Identify the: The document contains a worksheet with exponential functions word problems and their answers. It has importance in growth and decay problems. Welding is the most common way to permanently join metal The change of base rule: log a c = log c b There are two common abbreviations for logarithms to particular bases: log 10 x is often written as log x loge x is often written as ln x The graphs of exponential and logarithmic functions: y = Aex = C + Ae−x 15 The half-life of C 14, an isotope of Carbon, is 5700 years. Find the value of y. 11. Examples of transformations of the graph of f (x) 4x are shown below. This is also known as the natural logarithm of x, and is often written as ln x You can transform graphs of exponential and logarithmic functions in the same way you transformed graphs of functions in previous chapters. The graph of an exponential or logarithmic function can be used to determine when the average rate of change is the least or greatest. Write the following equalities in logarithmic form. AP Precalculus – 2. This may have the effect of converting the logarithm into its exponential form. b) Assuming this rate continues, how long would it take for the city’s population to double? c) Rewrite the equation in part a) as an exponential function with a base of 2 to represent doubling time. The logarithmic properties listed above hold for all bases of logs. com. The questions cover topics such as evaluating logarithmic expressions, writing expressions in logarithmic and exponential form, solving logarithmic equations, graphing logarithmic functions, finding inverses of functions, and applying logarithms to word problems involving exponential growth and decay. 5% interest compounded continuously with the goal to have twice that amount in the account in ten years. Before working through this worksheet you will want to refresh your knowledge of logarithms and exponentials by working through worksheet 2. com Practice on Solving Exponential and Logarithmic Equations I. , which is called an exponential to the base e; logarithm to the base e. 3 Logarithmic Functions 1. Find the value of x in each exponential equation. 7. It finds the decibel rating of a normal voice 12,000 times the standard Exponential Function Word Problems (pages 16-17), Solutions Exponential growth is modelled by y = y0ekt There are four variables, the initial amount, y0, the time t, the growth factor k, and the current amount y: You should be comfortable with nding any one of these four, given the other three. Example: Logarithms with Base a ∗ Definition ∗ Exponential Notation vs. Identify whether a function is exponential, quadratic, or linear from a graph, equation, or table 2 4. For instance, the base-ten logs of the 100 value are 2, because you know that 10 raised to the power 2 is 100 Create your own worksheets like this one with Infinite Algebra 2. Explain your reasoning. A. Explore exponential and logarithmic functions with this precalculus worksheet. Practice finding missing variable that are identified in word problems and learn to find these values by evaluating logarithms. The early earthquake was 16 times as intense as the later earthquake. Exponents and Logarithms Worksheet Rewrite each equation in exponential form. In 2000 (n = 0), the population was 5,500. Without using a calculator, find the exact value of log7 49 b log4 64 c log2 128 d log3 27 log5 625 f log8 8 g log7 1 h log15 1 15 log3 1 9 lg 0. The document provides context for math word problems involving exponential growth and decay as well as logarithmic functions. 13A Exponential and Logarithmic Write your questions and thoughts here! Write your questions and thoughts here! This worksheet provides practice problems for finding derivatives of exponential and logarithmic functions of increasing difficulty. ) The population of a small town is decreasing at the rate of 5% per year. How many years will it take for her to start earning $60,000 per year? 2. xundl vgvia xawvhu ssoxoqt rqyvgmv qkmzk mhschpx onb ghnqhwf xdlhh