Free ebook on exponents and logarithms for modeling growth and decay, graphing functions, and solving algebraic equations.
Free ebook content
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Exponential Expressions and Growth/Decay Thinking
+ Exercise: A value starts at 500 and keeps 92% of its value each year for n years. Which expression correctly models the value after n years?
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Exponent Rules as Algebra Tools
+ Exercise: Which expression is correctly simplified using exponent rules?
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Zero, Negative, and Fractional Exponents
+ Exercise: When working over the real numbers, what restriction on x is required to simplify x^{-1/2} / x^{3/2} into a form with only positive exponents?
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Exponential Functions and Their Graphs
+ Exercise: For the transformed exponential function f(x)=k·a^(x-h)+c, which statement correctly describes the horizontal asymptote and where the graph will lie relative to it when k>0?
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Building Exponential Models for Real Situations
+ Exercise: A savings account starts with $2,500 and grows by 0.6% per month. Which expression correctly models the balance after 3 years with time-step consistency?
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Why Logarithms: The Inverse of Exponentials
+ Exercise: Which statement correctly translates the logarithmic equation log_10(0.01) = -2 into exponential form?
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Logarithm Properties for Simplifying and Solving
+ Exercise: Which expression correctly condenses 3 log(x) - 2 log(y) + log(7) into a single logarithm (assuming all logs have the same base and are defined)?
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Solving Exponential Equations
+ Exercise: When solving an exponential inequality like (0.7)^t < 1/6 by taking natural logs, why does the inequality sign flip when you divide to solve for t?
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Solving Logarithmic Equations and Checking Validity
+ Exercise: When solving a logarithmic equation, why must you check candidate solutions in the original equation?
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Change of Base and Interpreting Log Scales
+ Exercise: On a base-b logarithmic scale, what does a difference of Δ between log_b(x2) and log_b(x1) represent in the original scale?
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Putting It Together: Growth and Decay Problem Solving
+ Exercise: A camera costs $1,600 new and loses 18% of its value each year. After solving the model, you find t ≈ 5.86 years for the value to reach $500. What is the best interpretation for “When will it be worth $500 or less?”
About the free ebook
Exponents and Logarithms: The Algebra You Need for Growth and Decay
This free ebook develops the algebraic ideas behind quantities that grow or shrink by repeated factors. Learn to read exponential expressions, apply exponent rules accurately, and work confidently with zero, negative, and fractional exponents.
Connect algebra to changing quantities
Exponential functions provide a practical way to describe repeated percentage change, including population growth, compound interest, radioactive decay, and depreciation. The ebook explains how to build models from real information and interpret their graphs, parameters, and predictions.
Use logarithms to solve for unknown exponents
Logarithms are introduced as the inverse operation of exponentiation. You will use logarithm properties to simplify expressions, solve exponential and logarithmic equations, and check that answers satisfy domain restrictions.
Skills you can apply
- Rewrite and simplify powers using exponent laws.
- Model growth and decay with exponential functions.
- Solve equations involving exponents and logarithms.
- Use change of base and interpret logarithmic scales.
- Choose meaningful answers for real-world problems.
Clear algebraic reasoning is emphasized throughout, helping students move from symbolic rules to reliable problem solving.
How do you solve an exponential equation when the bases are different?
Take logarithms of both sides, use log properties to isolate the exponent, and calculate the result.
Why must solutions to logarithmic equations be checked?
Every logarithm argument must be positive, so algebraic steps can produce invalid values that must be rejected.
What is change of base used for in logarithms?
It lets you evaluate a logarithm with common or natural logs when its original base is unavailable on a calculator.
This ebook includes:
11 content chapters
Digital certificate of course completion (Free)
Exercises to train your knowledge
100% free, from content to certificate
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