College Algebra

College Algebra covers fundamental algebraic concepts, including functions, equations, and inequalities, essential for higher-level mathematics.

Algebraic Expressions and OperationsSolving Equations and InequalitiesFunctions and Their Properties
Quadratic Equations and ApplicationsExponential and Logarithmic FunctionsSystems of Equations
Budgeting and Personal FinanceScience and Engineering ModelsBusiness and Economics
Practice and Review RegularlyBreak Down Complex Problems
Algebraic Expression...Solving Equations an...Functions and Their ...Quadratic Equations ...Exponential and Loga...Systems of EquationsBudgeting and Person...Science and Engineer...Business and Economi...Practice and Review ...Break Down Complex P...
Advanced Topics

Exponential and Logarithmic Functions

Going Beyond Linear Growth

Exponential functions grow rapidly (\(f(x) = a \cdot b^x\)), while logarithmic functions are their inverses (\(g(x) = \log_b(x)\)).

Exponential Functions

Used to model:

  • Population growth
  • Compound interest
  • Bacterial growth

Logarithmic Functions

Help reverse exponential growth, and are used in:

  • Decibel scales
  • Richter scale (earthquakes)

Key Properties

  • Exponential growth doubles, triples, etc., over regular intervals.
  • Logarithms “undo” exponentials: if \(2^x = 8\), then \(x = \log_2(8) = 3\).

Examples

  • If a population doubles every year, \(P(t) = 100 \cdot 2^t\) models its growth.

  • To solve \(3^x = 81\), rewrite as \(x = \log_3(81) = 4\).

In a Nutshell

Exponential and logarithmic functions model rapid change and their reversals, essential for science and finance.

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Systems of Equations