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Mathematics LibreTexts

4.6e: Exercises - Exponential and Logarithmic Equations

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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?

Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.

The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.

B: Solve Exponential Equations Using the 1-1 Property (like Bases)

Exercise \(\PageIndex{2}\) 

\( \bigstar \)  For the following exercises, use like bases to solve the exponential equation.

C: Solve exponential equations using logarithms

Exercise \(\PageIndex{3}\) 

\( \bigstar \)  For the following exercises, use logarithms to solve.

\( \bigstar \)  Solve. Give the exact answer and the approximate answer rounded to the nearest thousandth .

\( \bigstar \)  Find the \(x\)- and \(y\)-intercepts of the given function.

\( \bigstar \)  Use a \(u\)-substitution to solve the following.

D: Mixed exponential equations

Exercise \(\PageIndex{4}\) 

\( \bigstar \) S olve each exponential equation. Find the exact answer and then approximate it to three decimal places.

\( \bigstar \)  For the following exercises, solve the exponential equation exactly. 

\( \bigstar \)  For the following exercises, solve each equation. Write the exact solution, and then approximate the answer to \(3\) decimal places.

E: Solve log equations by rewriting in exponential form

Exercise \(\PageIndex{5}\) 

\( \bigstar \)  For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.

\( \bigstar \)  For the following exercises, use the definition of a logarithm to solve the equation.

\( \bigstar \)  Solve.

F: Solve log equations using the 1-1 property

Exercise \(\PageIndex{6}\) 

\( \bigstar \)  For the following exercises, use the one-to-one property of logarithms to solve.

G: Mixed log equations

Exercise \(\PageIndex{7}\) 

\( \bigstar \) S olve for \(x\). Give exact answer (not a decimal approximation).

\( \bigstar \) Solve for \(x\).

\( \bigstar \)  Solve for \(x\).

H: Inverses of Log and Exponent Functions

Exercise \(\PageIndex{8}\) 

\( \bigstar \)  Find the inverse of the following functions.

I: Mixed log and exponential equations

Exercise \(\PageIndex{9}\) 

J: Applications

Exercise \(\PageIndex{10}\) 

263. In chemistry, pH is a measure of acidity and is given by the formula \(\mathrm{pH}=-\log \left(H^{+}\right)\), where \(H^{+}\) is the hydrogen ion concentration (measured in moles of hydrogen per liter of solution.) Determine the hydrogen ion concentration if the pH of a solution is \(4\).

264. The volume of sound, \(L\) in decibels (dB), is given by the formula \(L=10 \log \left(I / 10^{-12}\right)\) where \(I\) represents the intensity of the sound in watts per square meter. Determine the intensity of an alarm that emits \(120\) dB of sound.

265. An account with an initial deposit of \(\$6,500\) earns \(7.25\%\) annual interest, compounded continuously. How much will the account be worth after \(20\) years?

266. The formula for measuring sound intensity in decibels \(D\) is defined by the equation \(D=10\log \left ( \frac{I}{I_0} \right )\) , where \(I\) is the intensity of the sound in watts per square meter and \(I_0=10^{-12}\) is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of \(8\cdot 3\cdot 10^2\) watts per square meter?

267. The population of a small town is modeled by the equation \(P=1650e^{0.5t}\) where \(t\) is measured in years. In approximately how many years will the town’s population reach \(20,000\)?

268. Atmospheric pressure \(P\) in pounds per square inch is represented by the formula \(P=14.7e^{-0.21x}\) , where \(x\) is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of \(8.369\) pounds per square inch? (Hint: there are \(5280\) feet in a mile)

269. The magnitude \(M\) of an earthquake is represented by the equation \(M=\dfrac{2}{3}\log \left ( \dfrac{E}{E_0} \right )\) where \(E\) is the amount of energy released by the earthquake in joules \(E_0=10^{4.4}\) and is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing \(1.4\cdot 10^{13}\) joules of energy?

270. Use the definition of a logarithm along with the one-to-one property of logarithms to prove that \(b^{\log_b x}=x\).

271. Recall the formula for continually compounding interest, \(y=Ae^{kt}\) .   Use the definition of a logarithm along with properties of logarithms to solve the formula for time \(t\) such that \(t\) is equal to a single logarithm.

272. Recall the compound interest formula \(A=a\left ( 1+\frac{r}{k} \right )^{kt}\) .   Use the definition of a logarithm along with properties of logarithms to solve the formula for time \(t\) .

273. Newton’s Law of Cooling states that the temperature \(T\) of an object at any time \(t\) can be described by the equation \(T=T_s+(T_0-T_s)e^{-kt}\) , where \(T_s\) is the temperature of the surrounding environment, \(T_0\) is the initial temperature of the object, and \(k\) is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time \(t\) such that \(t\) is equal to a single logarithm.

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solving exponential word problems with logarithms worksheet

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Logarithmic Word Problems

Log Probs Expo Growth Expo Decay

What are logarithm word problems?

Logarithmic word problems, in my experience, generally involve either evaluating a given logarithmic equation at a given point, or else solving an equation for a given variable; they're pretty straightforward.

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What real-world problems use logarithms?

The classic real-world contexts for logarithm word problems are the measurement of acidity or alkalinity (that is, the measurement of pH), the measurement of sound (in decibels, or dB), and the measurement of earthquake intensity (on the Richter scale), among other uses ( link ).

Note: While log-based word problems are, in my experience, pretty straightforward, their statements tend to be fairly lengthy. Expect to have to plow through an unusual amount of text before they get to the point.

  • Chemists define the acidity or alkalinity of a substance according to the formula pH =  −log[H + ] where [H + ] is the hydrogen ion concentration, measured in moles per liter. Solutions with a pH value of less than 7 are acidic; solutions with a pH value of greater than 7 are basic; solutions with a pH of 7 (such as pure water) are neutral.

a) Suppose that you test apple juice and find that the hydrogen ion concentration is [H + ] = 0.0003 . Find the pH value and determine whether the juice is basic or acidic.

b) You test some ammonia and determine the hydrogen ion concentration to be [H + ] = 1.3 × 10 −9 . Find the pH value and determine whether the ammonia is basic or acidic.

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In each case, I need to evaluate the pH function at the given value of [H + ] . In other words, this exercise, despite all the verbiage, is just plug-n-chug.

Since no base is specified, I will assume that the base for this logarithm is 10 , so that this is the so-called "common" log. (I happen to know that 10 is indeed the correct base, but they should have specified.)

a) In the case of the apple juice, the hydrogen ion concentration is [H + ] = 0.0003 , so:

pH = −log[H + ]

= −log[0.0003]

= 3.52287874528...

This value is less than 7 , so the apple juice is acidic.

b) In the case of the ammonia, the hydrogen ion concentration is [H + ] = 1.3 × 10 −9 , so:

= −log[1.3 × 10 −9 ] = 8.88605664769...

This value is more than 7 , so the ammonia is basic.

(a) The juice is acidic with a pH of about 3.5 , and (b) the ammonia is basic with a pH of about 8.9 .

When a logarithm is given without a base being specified, different people in different contexts will assume different bases; either 10 , 2 , or e . Ask now whether or not bases will be specified for all exercises, or if you're going to be expected to "just know" the bases for certain formulas, or if you're supposed to "just assume" that all logs without a specified base have a base of... [find out which one].

  • "Loudness" is measured in decibels (abbreviated as dB). The formula for the loudness of a sound is given by dB = 10×log[I ÷ I 0 ] where I 0 is the intensity of "threshold sound", or sound that can barely be perceived. Other sounds are defined in terms of how many times more intense they are than threshold sound. For instance, a cat's purr is about 316 times as intense as threshold sound, for a decibel rating of:

dB = 10×log[I ÷ I 0 ]     = 10×log[ (316 I 0 ) ÷ I 0 ]     = 10×log[ 316 ]     = 24.9968708262...

...about 25 decibels.

Considering that prolonged exposure to sounds above 85 decibels can cause hearing damage or loss, and considering that a gunshot from a .22 rimfire rifle has an intensity of about I = (2.5 × 10 13 )I 0 , should you follow the rules and wear ear protection when practicing at the rifle range?

I need to evaluate the decibel equation at I = (2.5 × 10 13 )I 0 :

dB = 10log [ I ÷ I 0 ]     = 10log[ (2.5 ×10 13 )I 0 ÷ I 0 ]     = 10log[2.5 ×10 13 ]     = 133.979400087...

In other words, the squirrel gun creates a noise level of about 134 decibels. Since this is well above the level at which I can suffer hearing damage,

I should follow the rules and wear ear protection.

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  • Earthquake intensity is measured by the Richter scale. The formula for the Richter rating of a given quake is given by R = log[ I ÷ I 0  ] where I 0 is the "threshold quake", or movement that can barely be detected, and the intensity I is given in terms of multiples of that threshold intensity.

You have a seismograph set up at home, and see that there was an event while you were out that had an intensity of I = 989I 0 . Given that a heavy truck rumbling by can cause a microquake with a Richter rating of 3 or 3.5 , and "moderate" quakes have a Richter rating of 4 or more, what was likely the event that occurred while you were out?

To determine the probable event, I need to convert the intensity of the mystery quake into a Richter rating by evaluating the Richter function at I = 989I 0 :

R = log[ I ÷ I 0 ]     = log[ 989I 0 ÷ I 0 ]     = log[989]     = 2.9951962916...

A Richter rating of about 3 is not high enough to have been a moderate quake.

The event was probably just a big truck.

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solving exponential word problems with logarithms worksheet

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Course: Algebra 1   >   Unit 12

  • Exponential expressions word problems (numerical)
  • Initial value & common ratio of exponential functions

Exponential expressions word problems (algebraic)

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Logarithm Word Problems Worksheets

The primary power of logarithms lies in their ability to help use model and understand exponential situations. This involves very large or very small values. 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 concentration. Logarithms also have a huge presence in the financial industries where it is all about watching or forecasting how our money will grow in the future. We have put together a good collection of word problems to help you explore how to tackle some common problems that people from these sectors are solving every day. We just put the scenarios in word problem form. These worksheets feature word problems that have a logarithmic aspect to them.

Aligned Standard: High School Modeling

  • Muni-Bonds Step-by-Step Lesson - Reed is trying to figure out how much his nest egg will be worth.
  • Guided Lesson - I would love to loan Michael money, if he'll pay it back at 14% interest.
  • Guided Lesson Explanation - The same formula is used for all the problems, just a slight adjustment in each one.
  • Practice Worksheet - A number of really good problems that take about seven to ten minutes each to solve.
  • Matching Worksheet - As my old professor would say, "These are problems for those that make the BIG BUCKS!" Hopefully our students will need to work on them for real.
  • Answer Keys - These are for all the unlocked materials above.
  • Homework Sheet 1 - Logs are used some much in the financial world, it's definitely a mainstay.
  • Homework Sheet 2 - You might just start to understand the power of a savings bond, when held for years.
  • Practice Worksheet 1 - You many notice that purposefully within the problems we refer to values in both word and numeric form to make them more real for students.
  • Practice Worksheet 2 - If you can find a bond that pays you that interest rate, keep it until it fully matures.
  • Quiz - Time to see how well you took to all of these problems.

How to Approach Logarithm Word Problems

Most students face the highest degree of difficulty when solving logarithmic word problems. Even though these word problems are much more straightforward, the kids find it complex to solve. The logarithmic word problems often will not make it clear which type of equation you need to use. You should work backwards and determine what they are looking for. Once you identify what is being asked, look around and see what is given. Then think ahead and see if there is an equation or formula that would allow you to use those items to find what they are looking for. All you have to do, from there, is enter the values in the equation to find the answer. This so much reminds of my entry-level college course. 1) Identify what they want. 2) Look at what is available. 3) See what formula fit all those aspects. 4) Plug in the number and bam, you have your solution.

Example: Let's say the neighbor's cat keeps getting in your garbage and you want to scare him away. How would you calculate the loudness of a cat's sound that is 316 decibels higher than the threshold sound. The formula to calculate loudness is dB = 10 log(I ÷ I o

Solution: To solve this word problem, all you have to do is insert the values in the formula.

dB = 10 log(316I o ÷ I o )

dB = 24.9968 decibels

Some word problems may require you to use logarithmic rules;

log b M = log n N --> M = N | log b M = N --> M = b N | log b M + log b N = log b (M.N) | log b M - log b N = log b (M/N)

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Word Problems With Exponents

Word Problems With Exponents - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Lesson 21 exponents and scientific notation, Exponent rules practice, Summer package pre requisite algebra skills, Word problem practice workbook, Work 2 7 logarithms and exponentials, Abeged mathematics activities student work, Sample work from, The product and power rules for exponents.

Found worksheet you are looking for? To download/print, click on pop-out icon or print icon to worksheet to print or download. Worksheet will open in a new window. You can & download or print using the browser document reader options.

1. Lesson 21: Exponents and Scientific Notation

2. exponent rules & practice, 3. summer package pre-requisite algebra skills, 4. word problem practice workbook, 5. worksheet 2 7 logarithms and exponentials, 6. abe/ged mathematics activities & student worksheets, 7. sample worksheet from www.mathmammoth, 8. the product and power rules for exponents.

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  1. 4.6e: Exercises

    Exercise 4.6e. 5. ★ For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. 121. log( 1 100) = − 2. 122. log324(18) = 1 2. ★ For the following exercises, use the definition of a logarithm to solve the equation. 123. 5log7n = 10.

  2. PDF Exponential and Logarithmic Equations and Applications

    1. Isolate the exponential expression on one side of the equation (if possible). 2. Take the log of both sides and "bring down the exponent" using the power property of logarithms. 3. Solve for the variable. RECALL: Properties of Logarithms For : = log (╽뀝) + log( 琩瀰) (╽뀝) − log 1. Solve. Leave your answer in 32 ) − 3 = 37 䫇惿≠ 1,╽뀝 > 0,琩瀰> 0,橿 ∈ RR

  3. PDF Exponential and Logarithmic Word Problems Notes Date

    Exponential and Logarithmic Word Problems Notes Date________________ Period____ Find the inverse of each function. 1) y = 1 3 + 6 -3 ) 2) y = log 5 (-4x + 6) + 4 3) y = 1 5 + 10 ) 2 4) y = ln (4x - 10) - 6 5) A substance decays 22% each day. After 7 days, there are 9 milligrams of the substance remaining.

  4. PDF Worksheet 2 7 Logarithms and Exponentials

    Logarithms and Exponentials Section 1 Logarithms The mathematics of logarithms and exponentials occurs naturally in many branches of science. It is very important in solving problems related to growth and decay. The growth and decay may be that of a plant or a population, a crystalline structure or money in the bank.

  5. PDF Math 3 Unit 9: Logarithms

    Purple Math: Solving Logarithmic Equations: Solving with Exponentials http://bit.ly/93elecba Patrick JMT: Solving Exponential Equations - Some Basic Examples http://bit.ly/93elecbb Patrick JMT: Properties of Logarithms - Everything You Need to Know! http://bit.ly/93elecbc Khan Academy: Logarithmic Equations: Variable in the Base

  6. Log problems: pH, decibels, and the Richter Scale

    The classic real-world contexts for logarithm word problems are the measurement of acidity or alkalinity (that is, the measurement of pH), the measurement of sound (in decibels, or dB), and the measurement of earthquake intensity (on the Richter scale), among other uses ( link ).

  7. PDF Solving Exponential Equations with Logarithms

    Solving Exponential Equations with Logarithms Date_____ Period____ Solve each equation. Round your answers to the nearest ten-thousandth. 1) 3 b = 17 2.5789 2) 12 r = 13 1.0322 3 ... Create your own worksheets like this one with Infinite Algebra 2. Free trial available at KutaSoftware.com. Title: Solving Exponential Equations with Logarithms

  8. Solving Exponential Equations Using Logarithms

    Steps to Solve Exponential Equations using Logarithms. 1) Keep the exponential expression by itself on one side of the equation. 2) Get the logarithms of both sides of the equation. You can use any bases for logs. 3) Solve for the variable. Keep the answer exact or give decimal approximations.

  9. Exponential Logarithmic Word Problems Teaching Resources

    This is a worksheet with 6 word problems. Students must write an exponential function to represent the scenario. They use logs to solve the functions. ... This worksheet provides practice solving exponential & logarithmic equations, including word problems that require the use of logarithms. There are 18 exponential equations to solve.

  10. Exponential model word problems (practice)

    Course: Algebra 2 > Unit 8 Math > Algebra 2 > Logarithms > Solving exponential models Exponential model word problems Google Classroom A culture of bacteria starts with 50 bacteria and increases exponentially. The relationship between B , the number of bacteria in the culture, and d , the elapsed time, in days, is modeled by the following equation.

  11. Exponential And Logarithm Word Problems Teaching Resources

    Browse exponential and logarithm word problems resources on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources. ... with the equations y = a(1+r)^t and y = a(1-r)^t. The student will solve for t, so logarithms should be used to solve. Upon completing the worksheet, the student will discover a ...

  12. Logarithms

    They allow us to solve challenging exponential equations, and they are a good excuse to dive deeper into the relationship between a function and its inverse. ... Solve exponential equations using logarithms: base-2 and other bases Get 3 of 4 questions to level up! Solving exponential models. Learn. Exponential model word problem: medication ...

  13. Exponential expressions word problems (algebraic)

    Exponential expressions word problems (algebraic) Google Classroom. Ngozi earns $ 24,000 in salary in the first year she works as an interpreter. Each year, she earns a 3.5 % raise. Write a function that gives Ngozi's salary S ( t) , in dollars, t years after she starts to work as an interpreter. Do not enter commas in your answer.

  14. Results for word problems exponential and logs

    We're Bruyn Math. This self checking worksheet contains 10 word problems dealing with the equations y = a (1+r)^t and y = a (1-r)^t. The student will solve for t, so logarithms should be used to solve. Upon completing the worksheet, the student will discover a Confucius-type thought. Subjects: Algebra 2, PreCalculus.

  15. Algebra

    Here are a set of practice problems for the Exponential and Logarithm Functions chapter of the Algebra notes. 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. At this time, I do not offer pdf's for solutions to individual ...

  16. PDF Worksheet: Logarithmic Function

    33x 1 2 x e (8) f(x) = 1 2e 2x 13. 15 000$ is invested in an account that yeilds 5% interest per year. After how many years will the account be worth 91 221.04$ if the interest is compounded yearly? 14. 8 000$ is invested in an account that yeilds 6% interest per year.

  17. PDF Solving Logarithmic Equations (Word Problems)

    Solving Logarithmic Equations (Word Problems) Example 1 INVESTMENT Mr. and Mrs. Mitchell are saving for their daughter's college education. They invest $10,000 in an account that pays 4.5% interest compounded continuously with the goal to have twice that amount in the account in ten years. a.

  18. PDF Exponential Equations Not Requiring Logarithms

    Create your own worksheets like this one with Infinite Algebra 2. Free trial available at KutaSoftware.com. ©X U2W0z1623 0KKuNtfaw PSkoTf4ttwFaDrpe4 lLqLnCn.e Y yAFlElJ WrEi4gihetQsB Urzes rePr5v1e6du.5 H oMQaSdIeN cwXiitBhN wIqnrf7iznYigtmeH cALlXg3eKbPrgal D2n.S.

  19. Logarithm Word Problems Worksheets

    Logarithm Word Problems Worksheets Click the buttons to print each worksheet and answer key. Logarithmic Expressions and Word Problems Lesson and Practice Students will practice using logarithmic expressions to answer word problems. example: William invested $2,244 in a six year CD that pays out ten percent compounded annually.

  20. Solving Word Problems Involving Applications of Exponential Functions

    Solving Word Problems Involving Applications of Exponential Functions to Growth and Decay. The population of a certain county can be modeled by the equation: Where is the population in millions, is the number of years since 1900. Find when the population is 100 million, 200 million, and 400 million. What do you notice about these time periods?

  21. Logarithm Word Problems Worksheets

    Logarithm Word Problems Worksheets. The primary power of logarithms lies in their ability to help use model and understand exponential situations. This involves very large or very small values. 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 ...

  22. Word Problems With Exponents Worksheets

    Word Problems With Exponents - Displaying top 8 worksheets found for this concept. Some of the worksheets for this concept are Lesson 21 exponents and scientific notation, Exponent rules practice, Summer package pre requisite algebra skills, Word problem practice workbook, Work 2 7 logarithms and exponentials, Abeged mathematics activities ...