F(x)=ab^{x} where b must be a positive real number

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Discussion Overview

The discussion revolves around the properties of the exponential function defined as f(x) = ab^x, specifically focusing on the requirement that the base b must be a positive real number. Participants explore the implications of using negative or complex bases.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants question why the base b of the exponential function must be positive and real, asking for reasons behind this restriction.
  • Others propose scenarios where b could be negative, such as b = -1, and inquire about the outcomes when x = 1/2.
  • One participant suggests that using a negative base leads to the concept of square roots of negative numbers, implying a connection to complex numbers.
  • Another participant confirms that using the imaginary unit i, which arises from the square root of -1, is not applicable in the context of the original discussion about real exponential functions.
  • There is acknowledgment that the discussion transitions into the realm of complex numbers when considering negative bases.

Areas of Agreement / Disagreement

Participants generally agree that the base of an exponential function should be positive and real, but there is contention regarding the implications and consequences of using negative bases, leading to a discussion about complex numbers.

Contextual Notes

The discussion does not resolve the mathematical implications of using negative bases in exponential functions, nor does it clarify the definitions of exponential functions in relation to complex numbers.

Avalance789
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Quote: "In mathematics, an exponential function is a function of the form

f(x)=ab^{x}

where b is a POSITIVE REAL number"

Wait. Give me a reason, why exponent base must be positive and real?

What happens if b<0?
 
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Avalance789 said:
Quote: "In mathematics, an exponential function is a function of the form

f(x)=ab^{x}

where b is a POSITIVE REAL number"

Wait. Give me a reason, why exponent base must be positive and real?

What happens if b<0?
What happens if b = -1 and x = 1/2?

-Dan
 
It means sqrt{-1}?

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Avalance789 said:
It means sqrt{-1}?

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Exactly. And as i is not real number we can't use it for what you seem to be working with.

-Dan
 
Ok, got it. So from this point we get into complex numbers.

Thank you!

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