Welch Teaching & Coaching
Trigonometry • College Preparatory Mathematics

Unit 6: Exponential & Logarithmic Functions

Connect exponential growth and decay to inverse logarithmic functions, equations, graphs, and models.

U6
Unit Road Map

Lesson Map

Use this table as your route through the unit. Lesson numbers jump directly to the matching explanation.

LessonTopicYou should be able to…
6.1Exponent PropertiesReview and apply integer and rational exponent rules.
6.2Exponential Equations: Same BaseRewrite expressions with a common base and equate exponents.
6.3Graphing ExponentialsCreate tables, identify intercepts, and describe end behavior.
6.4Growth & DecayInterpret exponential parameters in context.
6.5Inverse FunctionsDetermine invertibility, find inverses, and compare domain/range.
6.6Exponential ↔ Logarithmic FormTranslate between equivalent exponential and logarithmic statements.
6.7Solving With LogarithmsUse common logs and change of base to solve exponential equations.
6.8Properties of LogarithmsExpand, condense, and evaluate logarithmic expressions.
6.9Exponential ModelingCreate exponential equations from points and use models to predict outcomes.
Navigation

How the Unit Is Built

Each resource below is part of the same learning cycle. Select a card to jump to it.

Lesson MapSee the sequence and jump directly to any lesson.Content SummaryRead the ideas, methods, examples, and checks.Video LibraryOpen a targeted Khan Academy or YouTube search for each lesson.VocabularyReview the language needed to read and explain the mathematics.Sample ProblemsPractice with expandable worked solutions.Equations & RulesUse a compact reference for formulas and decision rules.Review GuidePrepare with a skill checklist and fresh mixed practice.Performance TaskApply the unit in a modeling or measurement context.
Learn

Content Summary

Work through these lessons in order. Focus on the meaning of the mathematics, the decision that selects a method, and a check that confirms the result.

Lesson 6.1

Exponent Properties

Review and apply integer and rational exponent rules.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 6.2

Exponential Equations: Same Base

Rewrite expressions with a common base and equate exponents.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 6.3

Graphing Exponentials

Create tables, identify intercepts, and describe end behavior.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 6.4

Growth & Decay

Interpret exponential parameters in context.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 6.5

Inverse Functions

Determine invertibility, find inverses, and compare domain/range.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 6.6

Exponential ↔ Logarithmic Form

Translate between equivalent exponential and logarithmic statements.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 6.7

Solving With Logarithms

Use common logs and change of base to solve exponential equations.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 6.8

Properties of Logarithms

Expand, condense, and evaluate logarithmic expressions.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 6.9

Exponential Modeling

Create exponential equations from points and use models to predict outcomes.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Watch & Reinforce

Video Library

These links open focused video searches by lesson topic. Use them as reinforcement rather than as a substitute for showing your own reasoning.

Lesson 6.1

Exponent Properties

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 6.2

Exponential Equations: Same Base

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 6.3

Graphing Exponentials

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 6.4

Growth & Decay

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 6.5

Inverse Functions

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 6.6

Exponential ↔ Logarithmic Form

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 6.7

Solving With Logarithms

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 6.8

Properties of Logarithms

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 6.9

Exponential Modeling

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Language

Vocabulary

These terms should appear naturally in explanations, diagrams, and written reasoning.

exponential function
Write a definition in your own words and add an example or sketch in your STRIDES notes.
growth factor
Write a definition in your own words and add an example or sketch in your STRIDES notes.
decay factor
Write a definition in your own words and add an example or sketch in your STRIDES notes.
inverse function
Write a definition in your own words and add an example or sketch in your STRIDES notes.
one-to-one
Write a definition in your own words and add an example or sketch in your STRIDES notes.
logarithm
Write a definition in your own words and add an example or sketch in your STRIDES notes.
base
Write a definition in your own words and add an example or sketch in your STRIDES notes.
change of base
Write a definition in your own words and add an example or sketch in your STRIDES notes.
domain
Write a definition in your own words and add an example or sketch in your STRIDES notes.
range
Write a definition in your own words and add an example or sketch in your STRIDES notes.
end behavior
Write a definition in your own words and add an example or sketch in your STRIDES notes.
Practice

Sample Problems

Try each problem before opening the solution. The examples are original practice aligned to the unit skills.

1

Same base

Solve 4^(x+1)=64.
Show solution
64=4³, so x+1=3 and x=2.
2

Growth

A quantity starts at 200 and grows 6% per period. Write a model.
Show solution
y=200(1.06)^t.
3

Inverse

Find the inverse of f(x)=2x−7.
Show solution
y=2x−7 → x=2y−7 → y=(x+7)/2.
4

Convert form

Rewrite 3⁴=81 in logarithmic form.
Show solution
log₃81=4.
5

Change of base

Write an expression for log₇20 using common logarithms.
Show solution
log20/log7.
6

Two-point model

A quantity is 50 at t=0 and 72 at t=2. Find a model y=ab^t.
Show solution
a=50; 72=50b², so b=√(72/50)=1.2. Model y=50(1.2)^t.
Reference

Equations & Rules

Know what each relationship means and when it applies; do not treat the box as a list of disconnected formulas.

Exponential Form
b^x=y
Log Form
log_b(y)=x
Change of Base
log_b(x)=log(x)/log(b)
Growth/Decay
y=a(b)^x
Inverse Check
f(g(x))=x and g(f(x))=x
Product Log
log_b(MN)=log_bM+log_bN
Quotient Log
log_b(M/N)=log_bM−log_bN
Power Log
log_b(M^p)=p log_bM
Prepare

Review Guide

Skill Checklist

  • I can review and apply integer and rational exponent rules.
  • I can rewrite expressions with a common base and equate exponents.
  • I can create tables, identify intercepts, and describe end behavior.
  • I can interpret exponential parameters in context.
  • I can determine invertibility, find inverses, and compare domain/range.
  • I can translate between equivalent exponential and logarithmic statements.
  • I can use common logs and change of base to solve exponential equations.
  • I can expand, condense, and evaluate logarithmic expressions.
  • I can create exponential equations from points and use models to predict outcomes.

How to review

  1. Complete the unit Skills Check packet without notes.
  2. Mark the exact skill behind each error.
  3. Return to that lesson and complete a fresh example.
  4. Explain the correction in words, then retry a similar problem.
  5. Finish with mixed practice so you must choose the method yourself.
Apply

Performance Task

Exponential Change Investigation

Choose a growth or decay situation, build an exponential model from data, justify the parameters, make a prediction, and explain when the model should and should not be trusted.

Evidence to include

A labeled mathematical model, organized calculations, appropriate units and precision, a written interpretation, and a check or discussion of limitations.