From Exponent Rules to Eigenvalues: A 132-Flashcard Study Guide to Logarithms, Complex Numbers and Matrices
Study 132 flashcards on exponent laws, logarithms, e, complex numbers, polar form, determinants, eigenvalues and Gaussian elimination in one deck.
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Four topics that look unrelated on a syllabus — exponent rules, logarithms, the imaginary unit and matrix algebra — turn out to be one continuous chain. Exponents, Logarithms, Complex Numbers & Matrices (code LY4423, 132 cards) on Flip It walks that chain from b² all the way to eigenvalues, which makes it a solid pre-calculus refresher or a first-year university survival deck.
Study this deck: Exponents, Logarithms, Complex Numbers & Matrices
1 / 10Exponentiation
Tap the card for the answer · swipe right if you knew it, left if not
Ten cards that show the range:
| Front | Back |
|---|---|
| Power of a power rule | (bᵐ)ⁿ = b^(mn) (multiply the exponents) |
| b^(m/n) | ⁿ√(bᵐ) or (ⁿ√b)ᵐ (nth root of b to the mth power) |
| How is 2^3^2 evaluated (order of operations)? | Right-associative (top-down): 2^(3²) = 2⁹ = 512, not (2³)² = 64 |
| Change of base formula | log_b(x) = log_c(x) / log_c(b) |
| Can you take the logarithm of zero? | No, log(0) is undefined (approaches negative infinity) |
| Multiply two complex numbers (a+bi) and (c+di) | (ac − bd) + (ad + bc)i, using i² = −1 |
| De Moivre's formula | (cos φ + i sin φ)ⁿ = cos(nφ) + i sin(nφ), used to compute powers of complex numbers |
| Determinant of a 2×2 matrix [[a,b],[c,d]] | det(A) = ad − bc |
| Eigenvalue and eigenvector | A scalar λ and nonzero vector v satisfying Av = λv |
| Cayley-Hamilton theorem | Every square matrix satisfies its own characteristic polynomial: p_A(A) = 0 |
What the 132 cards cover
- Exponent laws: b⁰ = 1, b¹, bᵐ·bⁿ, bᵐ/bⁿ, (bᵐ)ⁿ, (ab)ⁿ, (a/b)ⁿ, negative exponents, fractional exponents and nth roots, non-commutativity and non-associativity, the 0⁰ controversy, the binomial formula, even and odd functions.
- Powers in practice: scientific notation, the SI prefixes kilo, mega, giga and milli, 2¹⁰ = 1024, why a byte holds 2⁸ = 256 values, power sets with 2ⁿ subsets, compound interest A = P(1 + r/n)^nt, radioactive decay and exponential population growth.
- Logarithms: the definition as the inverse of exponentiation, product, quotient and power rules, change of base, log_b(1) = 0, log_b(b) = 1, common versus natural logs, e as lim(1 + 1/n)ⁿ, ln(e) = 1, pH, the Richter scale and decibels.
- Complex numbers: i² = −1, real and imaginary parts, the complex plane, conjugates, modulus |z| = √(a² + b²), argument, polar form, Euler's formula and Euler's identity, multiplication and division in polar form, n distinct nth roots, the fundamental theorem of algebra, complex logarithms, holomorphic functions and the Cauchy-Riemann equations.
- Matrices: m×n dimensions, square, row, column, diagonal, triangular and symmetric matrices, addition, scalar multiplication, transpose, the row-by-column product, non-commutativity, the identity matrix, inverses, determinants and det(AB) = det(A)det(B), rank, trace, orthogonal and positive-definite matrices, characteristic polynomials, LU decomposition, Gaussian elimination and the three elementary row operations.
Two habits that make this deck stick
Read every logarithm card backwards. log₂(8) = 3 is only memorable as "2³ = 8". Whenever a log card appears, restate it as an exponent statement before flipping; the product and power rules then stop being arbitrary, since they are just the exponent rules seen from the other side.
Collect the "no, it doesn't" cards. Exponentiation is neither commutative nor associative, matrix multiplication is not commutative, log(0) and logs of negatives are undefined over the reals, and complex exponentiation is multi-valued. These counterexamples are where most lost exam marks live — group them into one short review session of their own.
Search the Flip It library for deck LY4423 after signing in at flip-it.app. For the calculus that follows, see How to Learn Derivatives.
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