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mahmoud2011
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A Question about Defining Logarithms as integrals
- Context: Undergrad
- Thread starter mahmoud2011
- Start date
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- Tags
- Integrals Logarithms
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Discussion Overview
The discussion revolves around the definition of logarithms as integrals and the implications for exponential functions, particularly in relation to their properties and definitions in elementary algebra. Participants explore the relationship between logarithms and exponents, including rational exponents and their definitions as roots.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- Some participants question how defining logarithms as integrals affects the properties of exponential functions, particularly whether these functions align with those defined in elementary algebra.
- There is a discussion about the definition of rational exponents as roots, with some participants expressing uncertainty about how to establish this definition before understanding the properties of exponents.
- One participant proposes that if logarithms are defined as integrals, then basic properties such as L(xy) = L(x) + L(y) can be derived, but questions remain about the implications for rational exponents.
- Another participant suggests that understanding irrational exponents can simplify the understanding of rational exponents, but this raises further questions about the foundational definitions of exponents.
- Concerns are raised about the continuity and differentiability of the logarithm function and its implications for defining the exponential function.
Areas of Agreement / Disagreement
Participants express varying levels of clarity regarding the definitions and properties of logarithms and exponents. There is no consensus on how to approach the definitions of rational exponents or their relationship to logarithmic functions.
Contextual Notes
Participants highlight the need for a rigorous foundation in the properties of logarithms and exponents, noting that assumptions about continuity and differentiability are crucial for the discussion. There are unresolved questions about the definitions and existence of rational exponents prior to establishing their properties.
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