📐 Math ResourcesLast updated March 29, 2026

Algebra & Equations Guide: Exponents, Logarithms & Solving Techniques

Master algebraic concepts from basic equations to advanced functions

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Key Takeaways

Exponent rules: a^m × a^n = a^(m+n), (a^m)^n = a^(mn), a^0 = 1, a^(-n) = 1/a^n.
Logarithms are inverse exponents: log_b(x) = y means b^y = x. ln(x) = log_e(x).
To add fractions: find common denominator, add numerators: a/b + c/d = (ad + bc) / bd.
Quadratic formula solves ax² + bx + c = 0: x = (-b ± √(b² – 4ac)) / 2a.

Algebra is the language of mathematics — from simple equations to complex matrix operations. This guide covers fundamental and advanced algebraic concepts, drawing from Khan Academy and Purplemath curriculum standards — paired with our free equation solver and fraction calculator.

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Exponents & Powers

Exponents represent repeated multiplication: 2³ = 2×2×2 = 8. Key rules: product rule (a^m × a^n = a^(m+n)), power rule ((a^m)^n = a^(mn)), zero exponent (a^0 = 1), and negative exponents (a^(-n) = 1/a^n). Practice with our exponent calculator.

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Logarithms

Logarithms are inverse operations to exponentiation. If b^y = x, then log_b(x) = y. Common bases: log₁₀ (common log), ln = log_e (natural log), log₂ (binary/computing). Key properties: log(ab) = log(a) + log(b), log(a/b) = log(a) – log(b). Use our logarithm calculator for any base.

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Fractions & Equation Solving

Fraction operations require common denominators for addition/subtraction but not for multiplication/division. Our fraction calculator handles all operations with automatic simplification. For equations, our equation solver supports linear, quadratic, and polynomial equations.

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Matrices & Number Systems

Matrices support addition, subtraction, multiplication, transposition, determinants, and inverses. Use our matrix calculator for operations up to 4×4. Binary arithmetic (base 2) is fundamental to computing. Our binary calculator converts between binary, decimal, octal, and hexadecimal.

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How do I solve equations?

Isolate the variable by performing the same operation on both sides. For linear (ax + b = c): subtract b, divide by a. For quadratic (ax² + bx + c = 0): use the quadratic formula or factoring. Our equation solver handles these automatically.

What is a logarithm?

A logarithm answers "what exponent gives this result?" log₁₀(1000) = 3 because 10³ = 1000. Natural log (ln) uses base e ≈ 2.718. Logarithms are used in pH scales, decibels, Richter scale, and compound interest.

How do I multiply fractions?

Multiply numerators together and denominators together: (a/b) × (c/d) = (ac)/(bd). Then simplify by dividing both by their GCD. For division, multiply by the reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used in linear algebra, computer graphics, machine learning, and solving systems of equations. Our matrix calculator handles addition, multiplication, determinants, and inverses.

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