Math Problem Statement

ln(x - 1) =/= 0

Solution

The inequality ln(x1)0\ln(x - 1) \neq 0 implies that the natural logarithm of x1x - 1 must not equal zero. To analyze and solve this, we need to find the values of xx for which the natural logarithm is zero, and exclude them from the domain of x1x - 1.

Step-by-step Solution:

  1. The natural logarithm ln(y)\ln(y) is zero when y=1y = 1. Therefore: ln(x1)=0    x1=1    x=2\ln(x - 1) = 0 \implies x - 1 = 1 \implies x = 2

  2. The inequality ln(x1)0\ln(x - 1) \neq 0 means that x11x - 1 \neq 1, or equivalently: x2x \neq 2

  3. Domain Consideration: Since ln(x1)\ln(x - 1) is defined only when x1>0x - 1 > 0, it requires: x1>0    x>1x - 1 > 0 \implies x > 1

  4. Final Solution: Combining the above, the solution to ln(x1)0\ln(x - 1) \neq 0 is: x>1andx2x > 1 \quad \text{and} \quad x \neq 2 This can be written in interval notation as: x(1,2)(2,)x \in (1, 2) \cup (2, \infty)


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Related Questions:

  1. How do you solve ln(x1)=k\ln(x - 1) = k for xx when kk is a constant?
  2. What is the domain of ln(x1)\ln(x - 1) in general?
  3. How does the graph of ln(x1)\ln(x - 1) behave near x=1x = 1?
  4. What happens when x10x - 1 \leq 0 in ln(x1)\ln(x - 1)?
  5. How do you evaluate ln(ex)\ln(e^x) and simplify logarithmic expressions?

Tip:

Always remember that the natural logarithm ln(y)\ln(y) is defined only for y>0y > 0, 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