Precalculus and Trigonometry · 35 minutes
Suggested grades 11–12 · Unit 2: Exponential inverses and the unit circle · Lesson 1 of 2
Logarithms as Exponential Inverses
Translate exponential and logarithmic statements and solve a model on its real domain.
Mr. S Teaches · Original starter lesson · HS-PRE-03 · Grade range is planning guidance.Watch, notice, discuss
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Introduction to Logarithms
By Khan Academy
Ground the inverse relationship in simple exponent questions before using a logarithm to solve the model.
Open on YouTube: Introduction to Logarithms (opens a new tab) ↗Solving Exponential Equations
By The Organic Chemistry Tutor
Apply logarithms to unknown exponents and verify the result in the original equation.
Open on YouTube: Solving Exponential Equations (opens a new tab) ↗Graphing logarithmic functions | Exponential and logarithmic functions | Algebra II | Khan Academy
By Khan Academy
Compare the logarithmic graph with its exponential inverse and identify the positive-input restriction.
Open on YouTube: Graphing logarithmic functions | Exponential and logarithmic functions | Algebra II | Khan Academy (opens a new tab) ↗How does the exponential-logarithmic inverse relationship solve the equation, and which inputs are allowed?
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Learning objective
Solve exponential equations using the inverse relationship and explain input restrictions.
Before you begin
- Exponent laws
- Algebra II logarithmic solutions
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- logarithm base b
- The exponent on positive base b≠1 that produces a positive number.
- inverse relationship
- Two operations reversing one another on a suitable domain.
- domain restriction
- A condition required for an expression to have the intended real value.
Understand the concept
The statements b^y=x and log_b(x)=y express the same relationship when b>0, b≠1 and x>0. Logarithm laws come from exponent laws, but a log of a sum cannot be split into a sum of logs. Solving must retain positivity conditions; an algebraically manipulated candidate may not belong to the original domain.
Teach it step by step
- Translate 3⁴=81 into log₃81=4 and back.
- Solve 3^t=20 as t=ln 20/ln 3 and check approximately by exponentiation.
- Solve log₂(x−1)=3, state x>1 first and test the candidate.
- Compare log₂(4+4) with log₂4+log₂4 to expose the false sum rule.
Make the thinking visible
Worked example or model response
log₂(x−1)=3 gives x−1=8 and x=9, which satisfies x>1. log₂8=3 but log₂4+log₂4=4, so the supposed sum rule fails.
Guided practice
Solve log₅(2x+1)=2 and state the real input condition.
Independent practice
1. Solve 2^(3t)=7 using logarithms. 2. Solve log₃(x−4)=0. 3. Explain why log₂(−8) has no real value.
Quick assessment
What must be true of the logarithm’s argument before using a real-valued logarithm?
Answer key & assessment criteria
Answers and teaching guidance
- Guided:2x+1>0, hence x>−1/2; 2x+1=25 gives x=12.
- Independent 1: t=ln 7/(3ln2)≈0.936. 2: x−4=1, so x=5. 3: No real power of 2 is negative.
- Quick check: The argument must be positive.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Use a paired exponential/logarithm statement table and highlight the argument.
Extension challenge
Create a logarithmic equation whose manipulated candidate violates the original domain and explain the rejection.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSF.BF.B.5
Common Core · High school · Functions
(+) Additional preparation for advanced study
Use exponential/logarithmic inversion to solve and interpret equations.
Read the official standard ↗Additional preparation (+). The lesson focuses on simple real-valued equations, not all logarithmic identities or models.
Verified 2026-11-04. Selected contribution, not full attainment.