Solving exponential and logarithmic equations worksheet answer key

​Learning Objectives​

• Rewrite exponential and logarithmic equations into their alternate form to isolate a variable

• Solve exponential and logarithmic equations by applying the one-to-one property

• Combine knowledge of inverse operations and logarithm properties to solve logarithmic equation

​

Experience First

Today’s experience presents a learning progression from very simple to very complex logarithmic equations. First, students start by solving a quadratic equation to remind themselves of the key skills of combining like terms and inverse operations. Next, students reason about missing exponents and missing inputs of log equations and discover the value of rewriting an equation in its alternate form to isolate the question mark and make sense of the equation. In question 5, students must now complete at least one extra step to isolate their variable. In parts a and b, the extra step is to complete an inverse operation, in part c, the extra steps are to apply a log property and then complete an inverse operation.

Students were very successful on question 6 and again were asked to think about the idea of equivalence. In part d, they had to also apply a log property.

Question 7 is by far the most challenging, as students have to apply everything they’ve learned so far. If some groups don’t get to the end of this, that’s okay! Feel free to start the debrief when groups are still in a state of uncertainty about question 7, as this will increase engagement and heighten their desire for the new knowledge.

The goal of today’s lesson is to gather strategies for solving logarithmic and exponential equations, more so than to define a procedure for solving those equations. In the debrief, ask students to articulate why certain strategies were helpful and what the purpose of using that strategy is. 

​

While question 6 was easy for students to complete, the tie-in to the one-to-one property is more challenging, as students don’t readily see both sides of the equations as outputs of a function. To provide a non-example of the one-to-one property, ask students to think about y=x^2, a function that is NOT one-to-one. Ask them if knowing that the output is 9 would let them know for sure what the input is. When students say that the input could have been 3 or -3, ask them if this same scenario could occur with exponential or logarithmic functions. “Is it possible that two different inputs would have produced the same output?”

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To help students apply the one-to-one property, have them identify what was special or different about these equations from the equations in question 5. Students should be able to articulate that they saw exponential or logarithmic expressions with the same base on both sides of the equation. This is a defining characteristic that will help students on questions like 2b on the Check Your Understanding, where they should recognize that 8 can be written as 2^3 so both sides of the equation show the same base and thus the exponents can be set equal to each other.

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Solving Exponential Equations

An exponential equationAn equation which includes a variable as an exponent. is an equation that includes a variable as one of its exponents. In this section we describe two methods for solving exponential equations. First, recall that exponential functions defined by f(x)=bx where b>0 and b≠1, are one-to-one; each value in the range corresponds to exactly one element in the domain. Therefore, f(x)=f(y) implies x=y. The converse is true because f is a function. This leads to the very important one-to-one property of exponential functionsGiven b>0 and b≠1 we have bx=by if and only if x= y.:

bx=byifandonly ifx=y

Use this property to solve special exponential equations where each side can be written in terms of the same base.

Example 1

Solve: 32x−1=27.

Solution:

Begin by writing 27 as a power of 3.

32x−1=2732x−1=33

Next apply the one-to-one property of exponential functions. In other words, set the exponents equal to each other and then simplify.

2x−1=32x=4x=2

Answer: 2

Example 2

Solve: 161−3x=2.

Solution:

Begin by writing 16 as a power of 2 and then apply the power rule for exponents.

161−3x=2(24)1−3x=224(1−3x)=21

Now that the bases are the same we can set the exponents equal to each other and simplify.

4(1−3x)=14−12x=1−12x=−3x=−3−12=14

Answer: 14

Try this! Solve: 252x+3=125.

Answer: −34

In many cases we will not be able to equate the bases. For this reason we develop a second method for solving exponential equations. Consider the following equations:

32=93?=1233 =27

We can see that the solution to 3x=12 should be somewhere between 2 and 3. A graphical interpretation follows.

To solve this we make use of fact that logarithms are one-to-one functions. Given x,y>0 the one-to-one property of logarithmsGiven b>0 and b≠1 where x,y>0 we have logbx= logby if and only if x=y. follows:

logb x=logbyifandonlyifx=y

This property, as well as the properties of the logarithm, allows us to solve exponential equations. For example, to solve 3x=12 apply the common logarithm to both sides and then use the properties of the logarithm to isolate the variable.

3x=12log3x=log 12One-to-onepropertyoflogarithms xlog3=log12Powerruleforlogarith msx=log12log3

Approximating to four decimal places on a calculator.

x=log(12)/log(3)≈2.2619

An answer between 2 and 3 is what we expected. Certainly we can check by raising 3 to this power to verify that we obtain a good approximation of 12.

3^2.2618≈12✓

Note that we are not multiplying both sides by “log”; we are applying the one-to-one property of logarithmic functions — which is often expressed as “taking the log of both sides.” The general steps for solving exponential equations are outlined in the following example.

Example 3

Solve: 52x−1+2=9 .

Solution:

• Step 1: Isolate the exponential expression.

  52x−1+2=9 52x−1=7

• Step 2: Take the logarithm of both sides. In this case, we will take the common logarithm of both sides so that we can approximate our result on a calculator.

  log52x−1=log7

• Step 3: Apply the power rule for logarithms and then solve.

  log52x− 1=log7(2x−1)log5=log7 Distribute.2xlog5−log5=log72xlog5=log5+log7x=log5+log7 2log5

This is an irrational number which can be approximated using a calculator. Take care to group the numerator and the product in the denominator when entering this into your calculator. To do this, make use of the parenthesis buttons ( and ) :

x=(log 5+log(7))/(2*log(5))≈1.1045

Answer: log5+log 72log5≈1.1045

Example 4

Solve: e5x+3=1.

Solution:

The exponential function is already isolated and the base is e. Therefore, we choose to apply the natural logarithm to both sides.

e5x+3=1lne5x+3=ln1

Apply the power rule for logarithms and then simplify.

lne5x+3=ln1(5x+3)lne =ln1Recalllne=1andln1=0.(5x+3)⋅1=05x+3=0x =−35

Answer: −35

On most calculators there are only two logarithm buttons, the common logarithm LOG and the natural logarithm LN. If we want to approximate log3 10 we have to somehow change this base to 10 or e. The idea begins by rewriting the logarithmic function y=logax, in exponential form.

logax=y⇒x=ay

Here x>0 and so we can apply the one-to-one property of logarithms. Apply the logarithm base b to both sides of the function in exponential form.

x=ay logbx=logbay

And then solve for y.

logbx=ylogbalogb xlogba=y

Replace y into the original function and we have the very important change of base formulaloga x=logbxlogba; we can write any base-a logarithm in terms of base-b logarithms using this formula.:

log ax=logbxlogba

We can use this to approximate log310 as follows.

log310=log10log3≈2.0959 orlog310=ln10ln3≈2.0959

Notice that the result is independent of the choice of base. In words, we can approximate the logarithm of any given base on a calculator by dividing the logarithm of the argument by the logarithm of that given base.

Example 5

Approximate log7120 the nearest hundredth.

Solution:

Apply the change of base formula and use a calculator.

log7120=log120log7

On a calculator,

log(120)/log(7)≈2.46

Answer: 2.46

Try this! Solve: 23x+1−4=1. Give the exact and approximate answer rounded to four decimal places.

Answer: log5−log23 log2≈0.4406

Solving Logarithmic Equations

A logarithmic equationAn equation that involves a logarithm with a variable argument. is an equation that involves a logarithm with a variable argument. Some logarithmic equations can be solved using the one-to-one property of logarithms. This is true when a single logarithm with the same base can be obtained on both sides of the equal sign.

Example 6

Solve: log2(2x−5)−log2 (x−2)=0.

Solution:

We can obtain two equal logarithms base 2 by adding log2(x−2) to both sides of the equation.

log2(2x−5)−log2(x−2) =0log2(2x−5)=log2(x−2)

Here the bases are the same and so we can apply the one-to-one property and set the arguments equal to each other.

log2(2x −5)=log2(x−2)2x−5=x−2 x=3

Checking x=3 in the original equation:

log2(2(3)−5)=log2((3)−2) log21=log210=0 ✓

Answer: 3

When solving logarithmic equations the check is very important because extraneous solutions can be obtained. The properties of the logarithm only apply for values in the domain of the given logarithm. And when working with variable arguments, such as log(x−2), the value of x is not known until the end of this process. The logarithmic expression log(x−2) is only defined for values x>2.

Example 7

Solve: log(3x−4)=log(x−2) .

Solution:

Apply the one-to-one property of logarithms (set the arguments equal to each other) and then solve for x.

log(3x−4 )=log(x−2)3x−4=x−22x =2x=1

When performing the check we encounter a logarithm of a negative number:

log(x−2)=log(1−2)=log (−1)Undefined

Try this on a calculator, what does it say? Here x=1 is not in the domain of log(x−2). Therefore our only possible solution is extraneous and we conclude that there are no solutions to this equation.

Answer: No solution, Ø.

Caution: Solving logarithmic equations sometimes leads to extraneous solutions — we must check our answers.

Try this! Solve: ln(x2−15)−ln(2x)=0 .

Answer: 5

In many cases we will not be able to obtain two equal logarithms. To solve such equations we make use of the definition of the logarithm. If b>0, where b≠1 , then logbx=y implies that by=x. Consider the following common logarithmic equations (base 10),

logx=0⇒x=1Because100=1.logx= 0.5⇒x=?logx= 1⇒x=10Because101= 10.

We can see that the solution to logx=0.5 will be somewhere between 1 and 10. A graphical interpretation follows.

To find x we can apply the definition as follows.

log10x= 0.5⇒100.5=x

This can be approximated using a calculator,

x=10 0.5=10^0.5≈3.1623

An answer between 1 and 10 is what we expected. Check this on a calculator.

log3.1623≈5 ✓

Example 8

Solve: log3(2x−5)=2.

Solution:

Apply the definition of the logarithm.

log3(2x−5)=2 ⇒2x−5=32

Solve the resulting equation.

2x−5=92x=14x=7

Check.

log3(2(7)−5)=?2 log3(9)=2✓

Answer: 7

In order to apply the definition, we will need to rewrite logarithmic expressions as a single logarithm with coefficient 1.The general steps for solving logarithmic equations are outlined in the following example.

Example 9

Solve: log2( x−2)+log2(x−3)=1.

Solution:

• Step 1: Write all logarithmic expressions as a single logarithm with coefficient 1. In this case, apply the product rule for logarithms.

  log2(x−2)+log2(x−3)=1 log2[(x−2)(x−3)]=1

• Step 2: Use the definition and rewrite the logarithm in exponential form.

  log2[(x−2)(x−3)]=1 ⇒(x−2)(x−3)=21

• Step 3: Solve the resulting equation. Here we can solve by factoring.

  (x−2)(x−3)=2x2−5x+6=2x2−5x+4=0(x−4)(x−1)=0 x−4=0orx−1=0x=4x=1

• Step 4: Check. This step is required.

  Check x=4

  Check x=1

  log2(x−2)+log2(x−3)=1log2(4−2)+log2(4−3)=1log2(2)+log2(1)=11+0=1 ✓	log2(x−2)+log2(x−3)=1log2(1−2)+log2(1 −3)=1log2(−1)+log2(−2)=1✗

In this example, x=1 is not in the domain of the given logarithmic expression and is extraneous. The only solution is x=4.

Answer: 4

Example 10

Solve: log(x+15)−1=log(x+6).

Solution:

Begin by writing all logarithmic expressions on one side and constants on the other.

log(x+15)−1=log(x+6) log(x+15)−log(x+6)=1

Apply the quotient rule for logarithms as a means to obtain a single logarithm with coefficient 1.

log(x+15)−log(x+6)=1 log(x+15x+6)=1

This is a common logarithm; therefore use 10 as the base when applying the definition.

x+15x+6=101x+15 =10(x+6)x+15=10x+60−9x= 45x=−5

Check.

log (x+15)−1=log(x+6)log(−5+15)− 1=log(−5+6)log10−1=log11−1=00=0✓

Answer: −5

Try this! Solve: log2(x)+log2(x−1)=1.

Answer: 2

Example 11

Find the inverse: f(x)=log2(3x−4).

Solution:

Begin by replacing the function notation f(x) with y.

f(x)= log2(3x−4)y=log2(3x−4)

Interchange x and y and then solve for y.

x=log2(3y−4 )⇒3y−4=2x3 y=2x+4y=2x+43

The resulting function is the inverse of f. Present the answer using function notation.

Answer: f−1(x)=2x+43

Key Takeaways

• If each side of an exponential equation can be expressed using the same base, then equate the exponents and solve.
• To solve a general exponential equation, first isolate the exponential expression and then apply the appropriate logarithm to both sides. This allows us to use the properties of logarithms to solve for the variable.
• The change of base formula allows us to use a calculator to calculate logarithms. The logarithm of a number is equal to the common logarithm of the number divided by the common logarithm of the given base.
• If a single logarithm with the same base can be isolated on each side of an equation, then equate the arguments and solve.
• To solve a general logarithmic equation, first isolate the logarithm with coefficient 1 and then apply the definition. Solve the resulting equation.
• The steps for solving logarithmic equations sometimes produce extraneous solutions. Therefore, the check is required.

Topic Exercises

Part A: Solving Exponential Equations

Solve using the one-to-one property of exponential functions.

1. 3x=81

2. 2−x=16

3. 5x−1=25

4. 3x+4=27

5. 25x−2=16

6. 23x+7=8

7. 812x+1=3

8. 643 x−2=2

9. 92−3x−27=0

10. 81−5x−32=0

11. 16x2−2=0

12. 4x2−1−64=0

13. 9x(x +1)=81

14. 4x(2x+5)=64

15. 100x2−107x−3=0

16. e 3(3x2−1)−e=0

Solve. Give the exact answer and the approximate answer rounded to the nearest thousandth.

1. 3x=5

2. 7x=2

3. 4x=9

4. 2x=10

5. 5x− 3=13

6. 3x+5=17

7. 7 2x+5=2

8. 35x−9=11

9. 54x+3+6=4

10. 107x−1−2=1

11. e2x−3−5=0

12. e5x+1 −10=0

13. 63x+1−3=7

14. 8−109x+2=9

15. 15−e3x=2

16. 7+e4x+1=10

17. 7−9e−x=4

18. 3−6e−x=0

19. 5x2 =2

20. 32x2−x=1

21. 100e27x=50

22. 6e12x=2

23. 31+e−x=1

24. 21+3e−x=1

Find the x- and y-intercepts of the given function.

1. f(x)=3x+1−4

2. f(x)=23x−1−1

3. f(x)= 10x+1+2

4. f(x)=104x−5

5. f(x)=ex−2+1

6. f(x)=e x+4−4

Use a u-substitution to solve the following.

1. 32x−3x −6=0 (Hint: Let u=3x)

2. 22x+2x−20= 0

3. 102x+10x−12=0

4. 102x−10x−30=0

5. e2x−3ex+2= 0

6. e2x−8ex+15=0

Use the change of base formula to approximate the following to the nearest hundredth.

1. log25

2. log3 7

3. log5(23)

4. log7 (15)

5. log1/210

6. log2/330

7. log25

8. log263

9. If left unchecked, a new strain of flu virus can spread from a single person to others very quickly. The number of people affected can be modeled using the formula P(t)=e0.22t, where t represents the number of days the virus is allowed to spread unchecked. Estimate the number of days it will take 1,000 people to become infected.

10. The population of a certain small town is growing according to the function P(t)=12,500(1.02)t, where t represents time in years since the last census. Use the function to determine number of years it will take the population to grow to 25,000 people.

Part B: Solving Logarithmic Equations

Solve using the one-to-one property of logarithms.

1. log5(2x+ 4)=log5(3x−6)

2. log4(7x)= log4(5x+14)

3. log2(x−2)−log 2(6x−5)=0

4. ln(2x−1)=ln(3 x)

5. log(x+5)−log(2x+7)=0

6. ln(x2+4x)=2ln(x+1)

7. log32+2log3x=log3(7x−3)

8. 2logx−log36=0

9. ln(x+3)+ ln(x+1)=ln8

10. log5(x−2)+log 5(x−5)=log510

Solve.

1. log2(3x−7)=5

2. log3(2x+1)= 2

3. log(2x+20)=1

4. log4(3x+5)=12

5. log3x2=2

6. log(x2+3x+10)=1

7. ln(x2−1)=0

8. log5(x2+20)− 2=0

9. log2(x−5)+log2(x−9)= 5

10. log2(x+5)+log2(x+1)=5

11. log4x+log4(x−6)=2

12. log6x+log6(2x−1)=2

13. log3(2x+5)−log3(x−1)=2

14. log2(x+1)−log2(x−2)=4

15. lnx−ln(x−1)=1

16. ln(2x+1) −lnx=2

17. 2log3x=2+log3(2x−9)

18. 2log2x=3+log2(x−2)

19. log2(x−2)=2−log2x

20. log2(x+3)+log2(x+1)−1=0

21. logx−log(x+1)=1

22. log2(x+2) +log2(1−x)=1+log2(x+1)

Find the x- and y-intercepts of the given function.

1. f(x)=log(x+3)−1

2. f(x)=log(x−2)+1

3. f(x)=log 2(3x)−4

4. f(x)=log3(x+4) −3

5. f(x)=ln(2x+5)−6

6. f(x)=ln(x+1)+2

Find the inverse of the following functions.

1. f(x)=log2(x+5)

2. f(x)=4+log3x

3. f(x)=log(x+2 )−3

4. f(x)=ln(x−4)+1

5. f(x)=ln(9x−2)+5

6. f(x )=log6(2x+7)−1

7. g(x)=e3x

8. g(x)=10−2x

9. g (x)=2x+3

10. g(x)=32x+5

11. g(x)=10x+4−3

12. g(x)= e2x−1+1

Solve.

1. log(9x+5) =1+log(x−5)

2. 2+log2(x2+1)= log213

3. e5x−2−e3x=0

4. 3x2−11=70

5. 23x−5=0

6. log7(x+1)+log7(x−1)=1

7. ln(4x−1)−1=lnx

8. log (20x+1)=logx+2

9. 31+e2x=2

10. 2e−3x=4

11. 2e3x=e 4x+1

12. 2logx+logx−1=0

13. 3logx=log(x−2)+2logx

14. 2ln3+ lnx2=ln(x2+1)

15. In chemistry, pH is a measure of acidity and is given by the formula pH=−log (H+), 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.

16. The volume of sound, L in decibels (dB), is given by the formula L=10log(I/10−12) 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.

Part C: Discussion Board

1. Research and discuss the history and use of the slide rule.

2. Research and discuss real-world applications involving logarithms.

Answers

1. 4

3. 3

5. 6 5

7. −38

9. 16

11. ±12

13. −2, 1

15. 12, 3

17. log5log3≈1.465

19. log3log2≈1.585

21. 3log5+log13log5≈4.594

23. log2−5log 72log7≈−2.322

25. Ø

27. 3+ln52≈2.305

29. 1−log63log6 ≈0.095

31. ln133≈0.855

33. ln3≈1.099

35. ±log2 log5≈±0.656

37. −ln227≈−0.026

39. −ln2≈−0.693

41. x-intercept: (2log2−log3log3,0); y-intercept: ( 0,−1)

43. x-intercept: None; y-intercept: (0,12)

45. x-intercept: None; y-intercept: (0,1+e2e2)

47. 1

49. log3

51. 0, ln2

53. 2.32

55. −0.25

57. −3.32

59. 1.16

61. Approximately 31 days

1. 10

3. 35

5. −2

7. 12, 3

9. 1

11. 13

13. −5

15. ±3

17. ± 2

19. 13

21. 8

23. 2

25. ee−1

27. 9

29. 1+5

31. Ø

33. x-intercept: (7,0); y-intercept: ( 0,log3−1)

35. x-intercept: (163,0 ); y-intercept: None

37. x-intercept: (e6−52,0); y-intercept: (0,ln5−6)

39. f− 1(x)=2x−5

41. f−1(x)= 10x+3−2

43. f−1(x)=ex−5 +29

45. g−1(x)=lnx3

47. g−1(x)=log2x−3

49. g−1(x)=log(x+3)−4

51. 55

53. 1

55. log253

57. 14−e

59. ln(1/2)2

61. ln2−1

63. Ø

65. 10−4 moles per liter

1. Answer may vary

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