Domain of the Natural Logarithm Function (ln)
Key Concepts
- The natural logarithm function, denoted as ln, is only defined for positive values of x.
- For a function ( f(x) = \ln(x) ), it is only defined for ( x > 0 ).
- Extended forms, such as ( f(x) = \ln(x + a) ), require ( x + a > 0 ) to be valid.
Analyzing the Domain
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Basic ln Function
- For ( f(x) = \ln x ), the domain is ( x > 0 ).
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Shifted ln Function
- Example: ( f(x) = \ln(x + 2) )
- Domain: ( x + 2 > 0 ) which simplifies to ( x > -2 ).
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Reflected and Shifted ln Function
- Example: ( f(x) = \ln(-x + 4) )
- Domains calculated separately:
- ( -x + 4 > 0 ) which simplifies to ( x < 4 ).
- ( x + 2 > 0 ) which simplifies to ( x > -2 ).
- Combined valid range: ( -2 < x < 4 ).
-
Quadratic within ln
- Example: ( f(x) = \ln(x^2 + 8x + 12) )
- Solve ( x^2 + 8x + 12 > 0 ).
- Factorization gives roots at ( x = 2 ) and ( x = 6 ).
- Domain: ( x < 2 ) or ( x > 6 ).
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Complex Expression with ln
- Example: ( f(x) = \ln((x - 1)(x + 2)) )
- Conditions: ( (x - 1)(x + 2) > 0 ).
- Critical points at ( x = 1 ) and ( x = -2 ).
- Valid intervals: ( x > 1 ) or ( x < -2 ).
Additional Considerations
- Mixing multiple expressions within ln requires each to be positive for the function to be defined.
- When dealing with inequalities, solve for boundary values and determine valid intervals.
Examples and Exercises
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Basic Log Function
- ( f(x) = \ln(3x - 6) )
- Domain: ( x > 2 ).
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Single Variable ln
- ( f(x) = \ln(x) )
- Domain: ( x > 0 ).
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Quadratic and ln
- ( f(x) = \ln(x^2 - 5x + 24) )
- Factor and solve: ( (x - 8)(x + 3) = 0 )
- Domain: ( x > 8 ) or ( x < -3 ).
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Compound Expression with ln
- ( f(x) = \ln(x + 1) + \ln(2x + 6) )
- Solve: ( x > -3 ).
The above examples and analyses provide a comprehensive overview of determining the domain of functions involving natural logarithms, accommodating shifts, reflections, and more complex expressions.