HSF.BF.B.5: The Inverse Relationship Between Exponents and Logarithms
In plain English: HSF.BF.B.5 is an advanced (+) Common Core functions standard that asks students to understand logarithms as the inverses of exponents: log₂ 8 = 3 because 2³ = 8. Students use this relationship to switch between exponential and logarithmic form and to solve exponential and logarithmic equations, including growth and decay problems. It is usually taught in Algebra II or Precalculus.
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Common Core State Standards for Mathematics · Domain: Building Functions (BF) · Cluster: Build new functions from existing functions Also written as HSF-BF.B.5 or F-BF.5 · Official standard
A logarithm is an exponent. The statement log₂ 8 = 3 says the same thing as 2³ = 8: it names the power of 2 that gives 8. Students start from that sentence and rewrite facts in both directions until the two forms feel like one idea. They then see the relationship as a pair of inverse functions: f(x) = bˣ and g(x) = logb x undo each other, so logb(bˣ) = x and blogb x = x, and their graphs are mirror images across y = x.
The second half of the lesson puts the relationship to work. To solve an exponential equation such as 3 · 2ˣ = 45, students isolate the power and take a logarithm. To solve a logarithmic equation such as log₂(x + 3) = 5, they rewrite it in exponential form. Growth, decay, pH and decibel problems give the answers a meaning, and students decide how to report them in context.
Learning Objectives
By the end of this lesson, students will be able to:
Explain that logb x = y and bʸ = x express the same relationship, and rewrite statements between exponential and logarithmic form
Evaluate logarithms by finding the exponent of the base that gives the number
Use logb(bˣ) = x and blogb x = x to explain why bˣ and logb x are inverse functions, including their reflected graphs
Solve exponential equations by taking logarithms and logarithmic equations by rewriting in exponential form, checking that each solution is valid
Solve growth, decay and scale problems and interpret the answers in context
Prior Knowledge Required
Students should already be comfortable with:
Integer, negative and rational exponents HSN.RN.A.2
Inverse functions and the reflection of a graph across y = x HSF.BF.B.4
Exponential growth and decay models HSF.LE.A.2
Using the log and ln keys on a scientific calculator
Write four equations with a missing exponent on the board and ask students to fill in each one without a calculator.
Warm-Up Prompt
"Find the missing exponent: 2? = 64, 10? = 0.001, 5? = 1, 3? = 20. For which one can you only give an estimate? Between which two whole numbers is it?"
Collect the answers 6, -3 and 0. For 3? = 20, students should see that 3² = 9 and 3³ = 27, so the exponent is between 2 and 3. Explain that this exponent has a name, log₃ 20, and that the lesson is about how to find and use such exponents. Keep the question visible: students return to it with a calculator in Direct Instruction.
Direct Instruction25 minutes
Part 1: A logarithm is an exponent. Define logb x = y to mean bʸ = x, for b > 0, b ≠ 1 and x > 0. Read log₃ 20 aloud as "the exponent on 3 that gives 20." Introduce the two common bases: log x means log₁₀ x and ln x means logₑ x. On a calculator, log₃ 20 = ln 20/ln 3 ≈ 2.727, which settles the warm-up question. Use the top half of Diagram 2 to rewrite facts in both directions.
Part 2: Inverse functions. Show Diagram 1. Because 2³ = 8 puts (3, 8) on y = 2ˣ and log₂ 8 = 3 puts (8, 3) on y = log₂ x, every point of one graph is a point of the other with its coordinates swapped. So logb(bˣ) = x for every real x and blogb x = x for every x > 0. The range of bˣ, all positive numbers, is the domain of logb x, which is why the log of 0 or of a negative number is undefined.
Part 3: Solving. Use the bottom half of Diagram 2. Each equation is solved by applying the inverse of the operation that holds the variable:
Isolate the power bˣ or the logarithm logb(...) on one side of the equation.
Apply the inverse: take logb (or ln or log) of both sides of an exponential equation; rewrite a logarithmic equation in exponential form.
Simplify with the inverse property: logb(bᵘ) = u, or blogb u = u, removes the operation.
Solve the remaining equation, and give an exact answer before rounding.
Check by substitution, and make sure every logarithm in the original equation has a positive argument.
Rewriting and evaluating
Rewrite 4³ = 64 in logarithmic form. Then evaluate log₂(1/8).
Simplify 10log 7, log₅(52.4) and ln(e-3) without a calculator.
Equation: 7, 2.4 and -3: each function undoes the other
Exponential equation
Solve 3 · 2ˣ = 45. Divide by 3 to isolate the power, then take log₂ of both sides.
Equation: 2ˣ = 15, so x = log₂ 15 = ln 15/ln 2 ≈ 3.907
Logarithmic equation
Solve log₂(x + 3) = 5 by rewriting in exponential form.
Equation: x + 3 = 2⁵ = 32, so x = 29 (check: log₂ 32 = 5)
Growth in context
A bacteria culture starts with 500 cells and doubles every 3 hours: N(t) = 500 · 2t/3. When does it reach 12,000 cells?
Equation: 2t/3 = 24, so t = 3 log₂ 24 ≈ 13.75 hours, about 13 hours 45 minutes
After the last example, ask whether the answer should be rounded up or down if a lab checks the culture only on the hour. (The count passes 12,000 during the fourteenth hour, so the first hourly check above 12,000 is at 14 hours.)
Guided Practice15 minutes
Pairs solve five problems on whiteboards and hold them up after each one. (1) Rewrite 7² = 49 in logarithmic form (log₇ 49 = 2). (2) Rewrite log₃ 81 = 4 in exponential form (3⁴ = 81). (3) Solve 5ˣ = 30 (x = log₅ 30 ≈ 2.113). (4) Solve log x = 2.5 (x = 102.5 ≈ 316.2). (5) Solve ln x = 4 (x = e⁴ ≈ 54.60). Listen for these errors: dividing 30 by 5 in problem 3, reading log x = 2.5 as x = 2.5/10, and forgetting that ln means base e.
Independent Practice15 minutes
Students work alone on five problems and give exact answers first. (1) Evaluate log₆ 216 (3). (2) Solve e2x = 11 (x = ln 11/2 ≈ 1.199). (3) Solve 4 · 10ˣ = 900 (10ˣ = 225, x = log 225 ≈ 2.352). (4) Solve log₅(2x - 1) = 2 (x = 13). (5) Solve 2 ln x + 1 = 7 (ln x = 3, x = e³ ≈ 20.09). Students check problems 2 and 4 by substitution and say which inverse property they used in each one.
Closure5-10 minutes
Exit ticket: (1) Explain in one sentence why log₂(2⁹) = 9. (Answer: log₂ asks for the exponent on 2 that gives 2⁹, which is 9.) (2) Solve 6ˣ = 50. (Answer: x = log₆ 50 ≈ 2.183.) (3) Solve log₄ x = -1/2. (Answer: x = 4-1/2 = 1/2.)
Differentiation Strategies
For Struggling Students
Post a sentence frame on every desk: "logb x = y means b to the power y equals x", and have students say it aloud for each conversion
Start with powers of 2, 3 and 10 that give whole-number logarithms before any calculator work
Give a two-column "do and undo" chart: raising b to a power is undone by logb, and logb is undone by raising b to a power
For Advanced Students
Ask students to explain why logb x is undefined for b = 1 and for b < 0, using the graph of bˣ
Ask students to show that the solution of 5ˣ = 30 can be written as log 30/log 5 or ln 30/ln 5, and explain why both give the same number
Ask for the exact time at which two investments, 1000(1.06)ᵗ and 1500(1.03)ᵗ, have equal value, written using logarithms
Assessment Guidance
What to Look For
Listen for students describing a logarithm as an exponent, not as an operation to be multiplied: logb x = y should come with "so b to the y is x." In solving work, look for the power or the log isolated before the inverse is applied, an exact answer before a decimal one, and a check that every log argument is positive. In context problems, check that the reported answer fits the situation, for example a whole number of years when interest is paid once a year.
02
Classroom Activities
3 Activities
1
Two Forms, One Fact
15 minPairs
Pairs match 6 exponential-form cards with 6 logarithmic-form cards, then write a third card for each pair that says the fact in words. The goal is fluent conversion in both directions.
Shuffle all 12 cards face up; partners take turns placing a match and reading it aloud with the sentence frame
For each pair, write the words card, for example "the exponent on 8 that gives 4 is 2/3"
Sort the finished pairs into logarithms that are positive, zero and negative, and explain the pattern using the size of the number compared with 1
Challenge Variation
Give each pair two blank cards and ask them to write a matching pair whose logarithm is a negative fraction. Pairs trade and check each other's cards.
2
Mirror Graphs
20 minPairs
Pairs build the graph of y = log₃ x from the graph of y = 3ˣ by swapping coordinates, then use their graph to estimate logarithms and to see why the two functions are inverses.
Procedure
Make a table for y = 3ˣ at x = -2, -1, 0, 1, 2 (outputs 1/9, 1/3, 1, 3, 9) and plot it on graph paper
Swap each pair to get points of y = log₃ x, such as (9, 2) and (1/9, -2), plot them in a second color, and draw the line y = x
Use the log graph to estimate log₃ 5, then check with a calculator (ln 5/ln 3 ≈ 1.465)
Record the domain, range and asymptote of each graph in a two-column table
Discussion Questions
Where does each graph cross an axis, and why do those two points match?
Why can the log graph never reach x = 0 or cross to the left of it?
If you fold the paper along y = x, what happens to the two curves?
Modification for Distance Learning
Pairs use free graphing software: they enter y = 3ˣ, add a table, and type the swapped points as a second table before graphing y = log₃ x to compare.
3
Scenario Stations
25 minGroups of 3-4
Groups rotate through four stations. At each one they write an equation, solve it with the inverse relationship, and write one sentence that reports the answer in context with sensible rounding.
Station Cards
Savings: $2,000 earns 5% interest compounded once a year. After how many years is the balance first at least $3,000? (2000(1.05)ᵗ = 3000 gives t ≈ 8.31, so 9 years, because interest is added only once a year)
Sound: loudness in decibels is L = 10 log(I/I₀). How many times the reference intensity I₀ is an 85-decibel sound? (I/I₀ = 108.5 ≈ 316,000,000)
Acidity: pH = -log[H⁺]. Lemon juice has a pH of about 2.3. Find its hydrogen ion concentration. ([H⁺] = 10-2.3 ≈ 0.0050 moles per liter)
Medicine: 400 mg of a drug has a half-life of 6 hours, so 400(1/2)t/6 mg remain. When do 50 mg remain? ((1/2)t/6 = 1/8, so t = 18 hours)
Procedure
Spend about 6 minutes per station; roles rotate between writer, solver, checker and reporter
The checker substitutes the answer back into the original equation with a calculator
The reporter writes the context sentence and explains any rounding choice
Discussion Questions
At the savings station, why is 8.31 years not the answer to the question asked?
At which stations did you take a logarithm, and at which did you rewrite in exponential form?
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Diagrams & Visual Aids
2 diagrams
Diagram 1: y = 2ˣ and y = log₂ x Are Mirror Images
Graphs of y = 2ˣ and y = log₂ x, drawn to scale. Each point (a, b) on the exponential graph appears as (b, a) on the logarithmic graph, so the curves are reflections of each other across y = x. The domain and range trade places, and so do the asymptotes.
Diagram 2: Switching Forms and Undoing Each Operation
Top: bʸ = x and logb x = y state the same fact. Bottom: an exponential equation is solved by taking a logarithm, since ln(eᵘ) = u, and a logarithmic equation by rewriting it in exponential form. The last step checks that the log argument is positive.
04
Homework Assignment
~30 min
HSF.BF.B.5 Homework: Exponents and Logarithms
Directions: Show all work. Give an exact answer first, then a decimal rounded to three places where needed. Check each solution by substitution, and for word problems, answer in a complete sentence with units.
Part 1: The Inverse Relationship (Problems 1-2)
Rewrite each statement in the other form, then evaluate or complete it: (a) log₃ 243 = ? (b) log 0.0001 = ? (c) log₁₆ 4 = ? (d) 7-2 = 1/49.
Let f(x) = 4ˣ and g(x) = log₄ x. (a) Make a table of f for x = -1, 0, 1, 2, 3 and use it to write a table of g. (b) Evaluate 4log₄ 19 and log₄(4-3.5) without a calculator, and explain which property you used. (c) State the domain and range of f and of g, and explain how they are related.
Find and fix each error. (a) A student solves 2ˣ = 12 by writing x = 12/2 = 6. (b) A student solves log₃ x = 4 by writing x = 4³ = 64. For each, explain the mistake and give the correct solution.
Part 3: Problems in Context (Problems 5-6)
A town has 84,000 residents and grows by 2.5% per year, so P(t) = 84,000(1.025)ᵗ. (a) Write and solve an equation for when the population reaches 100,000. (b) In which year after the start does this happen? Check with P(7) and P(8).
The magnitude of an earthquake is M = log(A/A₀), where A is the amplitude of its seismic waves and A₀ is a fixed reference amplitude. (a) Rewrite the formula in exponential form to give A in terms of M. (b) How many times A₀ is the amplitude of a magnitude 5.8 earthquake? (c) How many times larger is the amplitude of a magnitude 6.3 earthquake than a magnitude 4.3 earthquake?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Converting Forms
Every statement rewritten correctly and evaluated
Most conversions correct
Forms confused
Inverse Properties
Properties used and explained, domain and range correct
Values correct but explanation missing
Missing or incorrect
Solving Equations
Exact and decimal answers, each checked, log arguments positive
Correct method with one error
Inverse not applied
Context
Answers interpreted with units and sensible rounding
Correct values, interpretation incomplete
No interpretation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which equation says the same thing as log₇ 343 = 3?
Answer: B
logb x = y means bʸ = x. Here b = 7, y = 3 and x = 343, so 7³ = 343, which is true: 7 · 7 · 7 = 343. Choice A swaps the base and the exponent, and choice C makes the number 343 the base.
Question 2 of 20 · Multiple Choice
Evaluate log₅ 125.
Answer: C
log₅ 125 asks for the exponent on 5 that gives 125. Since 5³ = 125, log₅ 125 = 3. Choice A divides 125 by 5, treating the log as division. Choice D has the wrong sign: 5-3 = 1/125.
Question 3 of 20 · Multiple Choice
Evaluate log₂(1/16).
Answer: A
1/16 = 2-4, so log₂(1/16) = -4. Choice B forgets that a number between 0 and 1 has a negative logarithm when the base is greater than 1. Choice C divides 16 by 2.
Question 4 of 20 · Multiple Choice
Simplify 10log 42.
Answer: D
log 42 is the exponent on 10 that gives 42, so raising 10 to that exponent gives 42. The functions 10ˣ and log x undo each other. Choice B stops after finding the exponent, and choice A multiplies 10 by 42.
Question 5 of 20 · Multiple Choice
Simplify ln(e5x).
Answer: B
ln is the logarithm with base e, so ln(eᵘ) = u for every real u. With u = 5x, the result is 5x. Choice A leaves the power in place, and choice C drops the variable.
Question 6 of 20 · Multiple Choice
The graph of y = log₃ x is the reflection of the graph of y = 3ˣ across which line?
Answer: A
log₃ x and 3ˣ are inverse functions, and the graph of an inverse is the reflection across y = x: each point (a, b) becomes (b, a). Choice C would give y = 3-x, and choice B would give y = -3ˣ.
Question 7 of 20 · Multiple Choice
The point (4, 81) is on the graph of y = 3ˣ. Which point must be on the graph of y = log₃ x?
Answer: C
3⁴ = 81 means log₃ 81 = 4, so (81, 4) is on the log graph: the coordinates swap. Choice D takes reciprocals, which is a different operation from finding an inverse function, and choice A changes a sign.
Question 8 of 20 · Multiple Choice
What is the domain of y = log₂ x?
Answer: D
The domain of log₂ x is the range of its inverse, 2ˣ, which takes only positive values. So log₂ x is defined only for x > 0. Choice B includes 0, but no power of 2 equals 0, so log₂ 0 is undefined.
Question 9 of 20 · Multiple Choice
Solve 4ˣ = 1024.
Answer: A
4⁵ = 1024, so x = log₄ 1024 = 5. Choice B divides 1024 by 4. Choice D uses the fact that 2¹⁰ = 1024, which gives the exponent for base 2, not base 4.
Question 10 of 20 · Multiple Choice
Solve 7ˣ = 60. Round to three decimal places.
Answer: D
Take a logarithm of both sides: x = log₇ 60 = ln 60/ln 7 ≈ 2.104. Check: 72.104 ≈ 60. Choice A divides instead of taking a logarithm, choice B uses base 10 instead of base 7, and choice C divides inside the logarithm instead of dividing two logarithms.
Question 11 of 20 · Multiple Choice
Solve log₄ x = 3.
Answer: C
Rewrite in exponential form: x = 4³ = 64. Check: log₄ 64 = 3. Choice A multiplies 4 by 3, and choice B computes 3⁴, swapping the base and the exponent.
Question 12 of 20 · Multiple Choice
Solve log(2x) = 3.
Answer: A
log means base 10, so 2x = 10³ = 1000 and x = 500. Check: log(1000) = 3. Choice B divides 3 by 2 as if log were a number multiplying 2x. Choice C stops at 2x = 1000, and choice D uses 10² instead of 10³.
Question 13 of 20 · Multiple Choice
A 200 mg dose of a medicine loses 20% of the remaining amount each hour, so 200(0.8)ᵗ mg remain after t hours. When are 50 mg left?
Answer: B
200(0.8)ᵗ = 50 gives 0.8ᵗ = 0.25, so t = ln 0.25/ln 0.8 ≈ 6.21 hours. Choice A divides the logarithms in the wrong order. Choice C assumes a constant loss of 40 mg per hour, which is linear decay, not exponential. Choice D stops at -ln 0.25 and never divides by ln 0.8.
Question 14 of 20 · Multiple Choice
Money in an account grows continuously at 4% per year, so it is multiplied by e0.04t after t years. When does it triple?
Answer: C
e0.04t = 3 gives 0.04t = ln 3, so t = ln 3/0.04 ≈ 27.47 years. Choice A divides 3 by 0.04 without taking a logarithm. Choice B multiplies ln 3 by 0.04 instead of dividing, and choice D is ln 3 alone.
Question 15 of 20 · Short Answer
Write 251/2 = 5 in logarithmic form. Then write log 100,000 = 5 in exponential form.
log₂₅ 5 = 1/2, because the exponent on 25 that gives 5 is 1/2. The base of log is 10, so the second statement is 10⁵ = 100,000.
Question 16 of 20 · Short Answer
Explain why logb(bˣ) = x and blogb x = x. Then use these facts to simplify 9log₉ 13 and log₁₂(120.7).
logb x is the exponent on b that gives x, so the functions bˣ and logb x are inverses and each one undoes the other. Raising b to the exponent that gives x returns x, and the exponent on b that gives bˣ is x. So 9log₉ 13 = 13 and log₁₂(120.7) = 0.7.
Question 17 of 20 · Short Answer
Solve 5 · 3ˣ + 2 = 67. Give the exact answer and a decimal rounded to three places.
Subtract 2 and divide by 5: 3ˣ = 13. Take a logarithm: x = log₃ 13 = ln 13/ln 3 ≈ 2.335. Check: 5 · 32.335 + 2 ≈ 5(13) + 2 = 67.
Question 18 of 20 · Short Answer
Solve log₃(2x + 5) = 4 and check your answer.
Rewrite in exponential form: 2x + 5 = 3⁴ = 81, so 2x = 76 and x = 38. Check: 2(38) + 5 = 81 > 0 and log₃ 81 = 4.
Question 19 of 20 · Short Answer
A phone bought for $900 loses 18% of its value each year, so its value after t years is 900(0.82)ᵗ dollars. When is it worth $300? In which year does that happen?
900(0.82)ᵗ = 300 gives 0.82ᵗ = 1/3, so t = ln(1/3)/ln 0.82 ≈ 5.54 years. After 5 years the phone is worth about $333.67 and after 6 years about $273.61, so its value drops to $300 during the sixth year.
Question 20 of 20 · Short Answer
Solve 10x - 1 = 250. Give the exact answer and a decimal rounded to three places.
Take log of both sides: x - 1 = log 250, so x = 1 + log 250 ≈ 3.398. Check: 102.398 ≈ 250.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.BF.B.5 mean?
HSF.BF.B.5 means students understand that logarithms and exponents are inverses and use that fact to solve problems. A logarithm is an exponent: log₁₀ 1000 = 3 because 10³ = 1000. Students switch between the two forms and solve exponential and logarithmic equations, including growth and decay problems in context.
Is HSF.BF.B.5 taught in Algebra 2 or Precalculus?
It is usually taught in Algebra II, and many Precalculus courses revisit it. The standard is marked (+), which Common Core uses for additional mathematics that students should learn to take advanced courses such as calculus.
What is the inverse relationship between exponents and logarithms?
The functions f(x) = bˣ and g(x) = logb x undo each other. Taking logb of bˣ gives back x, and raising b to the power logb x gives back x. Their graphs are reflections of each other across the line y = x, and the domain of one is the range of the other.
How do you convert between exponential and logarithmic form?
Use the sentence "logb x = y means b to the power y equals x." The base stays the base, and the logarithm is the exponent.
2⁵ = 32 becomes log₂ 32 = 5
log₃ 81 = 4 becomes 3⁴ = 81
ln x = 2 becomes e² = x
How do you solve an exponential equation like 3ˣ = 20?
Take a logarithm of both sides. The exact answer is x = log₃ 20. To get a decimal, use ln 20/ln 3 or log 20/log 3, both about 2.727. If the power has a coefficient, as in 4 · 3ˣ = 80, divide first so the power is alone.
How do you solve a logarithmic equation?
Isolate the logarithm, then rewrite the equation in exponential form. For log₂(x - 1) = 6, write x - 1 = 2⁶ = 64, so x = 65. Always check that the expression inside every logarithm is positive for your answer, because the log of zero or a negative number is undefined.
Why can't you take the logarithm of 0 or a negative number?
Because no power of a positive base gives 0 or a negative number. The logarithm asks "which exponent on b gives x?", and for x ≤ 0 there is no such exponent. On the graph, bˣ stays above the x-axis, so its inverse is only defined for x > 0.
What is the difference between log and ln?
They have different bases. log x usually means the base-10 logarithm, and ln x is the natural logarithm with base e ≈ 2.718. Both are inverses of exponential functions: log undoes 10ˣ and ln undoes eˣ. Either key can be used to evaluate a logarithm with another base, such as log₅ 40 = ln 40/ln 5.
What are common mistakes with logarithms?
A frequent error is treating log as a number that multiplies, so that log(2x) = 3 is "solved" by dividing 3 by 2. Others include dividing to solve 2ˣ = 12 (x = 6 is wrong), swapping the base and exponent when rewriting log₃ x = 4 as x = 4³, and forgetting to check that the log arguments are positive.
How does HSF.BF.B.5 connect to other standards?
It builds on inverse functions from HSF.BF.B.4. In HSF.LE.A.4, students write the solution of an exponential model such as abct = d as a logarithm and evaluate it with technology, which uses the same relationship. Later, the idea of solving an equation by applying an inverse function returns with inverse trigonometric functions (HSF.TF.B.7) and in calculus.
07
Related Standards
5 standards
These standards connect to HSF.BF.B.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSF.BF.B.4Prerequisite
Find inverse functions, including by solving f(x) = c
