Lesson 9 of 12 · Algebra Foundations
Exponents, Radicals & Logarithms
Three topics that are really one: a logarithm is just an exponent asking a question, and a radical is just a fractional exponent in disguise. Fluency here means converting freely between all three forms without hesitation.
Mark your status for this lesson:
Learning objectives
- Apply the exponent rules — product, quotient, power of a power, negative and fractional exponents — fluently.
- Simplify and combine radical expressions by factoring out perfect squares.
- Convert between exponential and logarithmic form, and apply the product, quotient, and power rules for logarithms.
- Solve exponential equations by matching bases, and logarithmic equations by converting to exponential form.
The core idea
Every exponent rule follows from the same source: aᵐ · aⁿ = aᵐ⁺ⁿ counts how many factors of a you have. Radicals are just fractional exponents — √a = a^(1/2) — so every radical simplification is secretly an exponent simplification. And a logarithm answers "what exponent gives this result?" — log_b(x) = y means bʸ = x, exactly. Once you see all three as one idea, the rules stop being separate things to memorize.
Exponent rules
aᵐ·aⁿ=aᵐ⁺ⁿ, aᵐ/aⁿ=aᵐ⁻ⁿ, (aᵐ)ⁿ=aᵐⁿ
a⁰=1, a⁻ⁿ=1/aⁿ, aᵐ/ⁿ=(ⁿ√a)ᵐ
Simplifying and combining radicals
Factor out the largest perfect square from the radicand: √(k²b) = k√b. Radicals combine like ordinary terms only when the part under the root matches — 5√2 + 3√2 = 8√2, but 5√2 + 3√3 doesn't simplify further.
Logarithm definition and rules
log_b(x) = y ⟺ bʸ = x
log_b(xy)=log_b x+log_b y, log_b(x/y)=log_b x−log_b y, log_b(xⁿ)=n·log_b x
Worked example 1 — Combining exponent rules
Problem
Simplify (2³ × 2⁵)/2⁴ to a single number.
Key insight
Combine all the exponents on the common base 2 before computing anything.
Solution
2³⁺⁵⁻⁴ = 2⁴ = 16.
Takeaway
Never compute each power separately when the base matches — combine the exponents first, then evaluate once.
Worked example 2 — Simplifying a radical
Problem
Simplify √72.
Key insight
Find the largest perfect square that divides 72.
Solution
√72 = √(36×2) = 6√2.
Takeaway
Always check for the largest perfect square factor — using a smaller one (like 4) works but leaves an extra simplification step.
Worked example 3 — Solving an exponential equation
Problem
Solve 3^(2x−1) = 27.
Key insight
Write both sides as powers of the same base, then match the exponents.
Solution
27 = 3³ → 2x−1=3 → x=2.
Takeaway
Whenever both sides of an exponential equation can be written with the same base, the exponents themselves must be equal.
Worked example 4 — Solving a logarithmic equation
Problem
Solve log₂(x+3) = 4.
Key insight
Convert directly to exponential form using the definition of a logarithm.
Solution
x+3 = 2⁴ = 16 → x = 13.
Takeaway
"log_b(expr) = y" converts directly to "expr = bʸ" — no other algebra needed to clear the logarithm.
Strategy notes
Match bases before matching exponents
Almost every exponential equation on the AMC 10 can be rewritten so both sides share a common base (often 2 or 3). Once that's done, the equation collapses to a simple linear one.
Convert logs to exponential form immediately
A logarithmic equation is almost always easier to solve the moment you rewrite it in exponential form — resist the temptation to manipulate it as a logarithm any longer than necessary.
Common mistakes
Combining radicals with different radicands
5√2 + 3√3 does not simplify to 8√6 or anything else — radicals only combine like ordinary like terms, which requires the same number under the root.
Forgetting to check the domain of a logarithmic equation
A logarithm is only defined for a positive argument. After solving, check that every root still keeps every logarithm's argument positive — reject any that don't.
Misapplying the power rule to a sum
log_b(x + y) is not log_b(x) + log_b(y) — the product rule applies to log_b(xy), a completely different expression. Don't confuse the two.
Practice
20 questions across four difficulty tiers. Most want a single number. Radical-simplification questions ask for the form a√b — type your answer as two comma-separated values, a then b. Up to three tries per question before the solution is shown.