Defining the Range of Variables in Logarithmic and Radical Expressions

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SUMMARY

The discussion focuses on determining the range of values for variables x and y in the equation ln(2+y) = 5ln(3 - x) - 2√x. It is established that the natural logarithm function ln(γ) is defined for γ > 0, and the square root function √η is defined for η ≥ 0. Consequently, the conditions for x and y are derived as follows: 2 + y > 0, 3 - x > 0, and x ≥ 0. These constraints lead to the conclusion that y must be greater than -2, x must be less than 3, and x must be non-negative.

PREREQUISITES
  • Understanding of logarithmic functions and their domains
  • Knowledge of square root functions and their domains
  • Familiarity with solving inequalities
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of logarithmic functions, specifically their domains and ranges
  • Learn about the characteristics of square root functions and their implications in inequalities
  • Explore methods for solving inequalities involving multiple variables
  • Practice problems involving logarithmic and radical expressions to solidify understanding
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Students studying algebra, particularly those focusing on logarithmic and radical expressions, as well as educators looking for examples to illustrate these concepts.

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Homework Statement



Real variables x and y are related by the equation

ln(2+y) = 5ln(3 - x) - 2 sqrt x ...(sorry, I haven't as yet got the hang on LaTeX)

Determine the range of values of x and y for which the expressions on each side of this equation are defined.


Homework Equations





The Attempt at a Solution



I haven't really been able to make an attempt at a solution. I think I have to take the exp of each side, but I am not sure excatly what way I go about this, so if someone could give me some advice, that would be fantastic.

Thanks in advance.

Sean
 
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HINTS:

(1) For what values of [itex]\gamma[/itex] is [itex]\ln\gamma[/itex] defined?

(2) For what values of [itex]\eta[/itex] is [itex]\sqrt{\eta}[/itex] defined?
 

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