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Properties of Logarithms definitions

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  • Logarithm
    Represents the exponent needed for a base to produce a given number; connects multiplication, division, and powers to addition, subtraction, and multiplication.
  • Base
    The fixed number in a logarithmic expression that is raised to a power to yield the argument.
  • Argument
    The value inside a logarithmic function, representing the number for which the logarithm is calculated.
  • Product Property
    Allows rewriting a log of a product as the sum of logs with the same base, turning multiplication inside into addition outside.
  • Quotient Property
    Allows rewriting a log of a quotient as the difference of logs with the same base, turning division inside into subtraction outside.
  • Power Property
    Allows moving an exponent in the argument to the front as a coefficient, or vice versa, connecting powers and multiplication.
  • Expansion
    The process of rewriting a single logarithm as a sum, difference, or multiple of several logarithms using log properties.
  • Condensation
    The process of combining multiple logarithmic terms into a single logarithm by applying log properties in reverse.
  • Exponent
    Indicates how many times the base is multiplied by itself; can be moved in or out of a logarithm using the power property.
  • Coefficient
    A number multiplying a logarithm, which can be moved into the argument as an exponent using the power property.
  • Sum
    The result of adding two or more logarithms, often produced by expanding a log of a product.
  • Difference
    The result of subtracting one logarithm from another, often produced by expanding a log of a quotient.
  • Same Base
    A requirement for combining or condensing logarithms using product or quotient properties; all logs involved must share this feature.
  • Multiplication
    An operation inside a logarithm that, when present, allows use of the product property to expand into a sum of logs.
  • Division
    An operation inside a logarithm that, when present, allows use of the quotient property to expand into a difference of logs.
  • Addition
    An operation inside a logarithm that cannot be simplified using log properties; does not correspond to a log property.
  • Subtraction
    An operation inside a logarithm that cannot be simplified using log properties; does not correspond to a log property.
  • Root
    A radical expression in the argument, which can be rewritten as a fractional exponent to apply the power property.
  • Fractional Exponent
    An exponent expressed as a fraction, often used to represent roots within logarithmic arguments.
  • Negative Exponent
    An exponent less than zero, indicating reciprocal values; useful for rewriting fractions in logarithmic arguments.
  • Single Logarithm
    A logarithmic expression not split into a sum or difference; often the result of condensing multiple logs.
  • Multiple Logarithms
    A sum or difference of two or more logarithmic terms, often produced by expanding a single log.
  • Simplification
    The process of rewriting logarithmic expressions in a more compact or expanded form using log properties.