A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Functions
CSCA Mathematics · Lesson 5 of 16Exponents and logarithms
Puissances et logarithmes · القوى واللوغاريتمات
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Exponents describe repeated multiplication; logarithms answer the question “to what power?”. They appear in growth problems and equations.
Key ideas
- aᵐ · aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1.
- log_a(x) = y means aʸ = x.
- log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n·log x.
- To solve aˣ = b, take logarithms of both sides.
Formulas to know
- a⁻ⁿ = 1/aⁿ
- log_a(a) = 1, log_a(1) = 0
- Change of base: log_a(x) = ln x / ln a
Worked example
Solve 2ˣ⁺¹ = 32.
- Write 32 as a power of 2: 32 = 2⁵.
- So 2ˣ⁺¹ = 2⁵, which means x + 1 = 5.
Answer: x = 4
Practice questions
1. Simplify 2³ · 2⁴.
2⁷ ✓ · 2¹² · 4⁷ · 2¹
Add the exponents: 3 + 4 = 7.
2. log₃(81) = ?
3 · 4 ✓ · 9 · 27
3⁴ = 81.
3. log 2 + log 5 = ? (base 10)
log 7 · 1 ✓ · 10 · 0.7
log 2 + log 5 = log 10 = 1.