Math Problem Statement
ln(x - 1) =/= 0
Solution
The inequality implies that the natural logarithm of must not equal zero. To analyze and solve this, we need to find the values of for which the natural logarithm is zero, and exclude them from the domain of .
Step-by-step Solution:
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The natural logarithm is zero when . Therefore:
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The inequality means that , or equivalently:
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Domain Consideration: Since is defined only when , it requires:
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Final Solution: Combining the above, the solution to is: This can be written in interval notation as:
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Related Questions:
- How do you solve for when is a constant?
- What is the domain of in general?
- How does the graph of behave near ?
- What happens when in ?
- How do you evaluate and simplify logarithmic expressions?
Tip:
Always remember that the natural logarithm is defined only for , so checking the domain of the argument is essential before solving logarithmic equations or inequalities.
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Math Problem Analysis
Mathematical Concepts
Logarithmic Inequalities
Natural Logarithm
Domain of Functions
Formulas
ln(y) = 0 if and only if y = 1
ln(x - 1) is defined for x - 1 > 0
Theorems
Properties of Logarithms
Domain Restrictions of Logarithmic Functions
Suitable Grade Level
Grades 10-12