Log base 2 is the same thing as square root?

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Discussion Overview

The discussion revolves around the relationship between logarithms, specifically logarithm base 2, and square roots. Participants explore whether these two mathematical concepts can be considered equivalent or related in any way.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions if the logarithm base 2 of a number is the same as the square root of that number.
  • Another participant suggests testing various values to see if the two yield the same results.
  • A participant clarifies that logarithms and square roots represent different mathematical operations, explaining that logarithms are inverses of exponentiation while square roots relate to squaring a number.
  • There is a mention of a relationship involving the natural logarithm and square roots, specifically that ##\sqrt{x}= e^\frac{\ln{x}}{2}##.
  • Participants agree on the definitions of logarithms, stating that ##\log_{b} x = y## implies that ##b^y=x##.
  • One participant emphasizes that the two equations are equivalent, meaning they share the same solutions and graphs.

Areas of Agreement / Disagreement

Participants generally agree on the definitions of logarithms and their equivalence to exponential forms, but there is disagreement regarding the initial claim that logarithm base 2 and square roots are the same.

Contextual Notes

The discussion does not resolve the initial question about the equivalence of logarithms and square roots, and it highlights the need for careful consideration of definitions and operations.

Who May Find This Useful

This discussion may be useful for individuals interested in understanding the properties of logarithms and square roots, as well as those exploring foundational concepts in mathematics.

xeon123
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Hi,

Is is correct to say that the logarithm of base 2 of a number x, is the same thing as the square root of a number x?
 
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Have you tried it on some values? Do you get the same results?
 
No, not at all.

To say that ##\log_2{x} = y## you mean that ##x=2^y##, logarithms are just ways of 'inversing' exponentiation (roughly). To say that ##\sqrt{x}=y## you are saying that ##x = y^2##, completely different.

However, there is a neat little tidbit that says that ##\sqrt{x}= e^\frac{\ln{x}}{2}##
 
Ok. But I can say that these 2 expressions are correct?

\log_{b} x = y, and b^y=x
 
Last edited:
Yes, just the definition of logs.
 
xeon123 said:
Ok. But I can say that these 2 expressions are correct?

\log_{b} x = y, and b^y=x

The appropriate terminology is that the two equations are equivalent. This means that any ordered pair (x, y) that satisfies one equation also satisfies the other. It also means that both equations have the same graph.
 

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