Logarithm inequality divide an inequality by a negative value

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SUMMARY

The discussion centers on the mathematical principle that when dividing an inequality by a negative value, the direction of the inequality must be reversed. Specifically, the example provided illustrates that since $$\log_{10}\left(\frac{1}{2}\right)<0$$, dividing both sides of an inequality by this negative logarithmic value results in the reversed inequality, leading to the conclusion that $$2<3$$. This highlights a fundamental rule in inequality manipulation that is crucial for accurate mathematical reasoning.

PREREQUISITES
  • Understanding of logarithmic functions, specifically base 10 logarithms.
  • Familiarity with basic inequality properties in mathematics.
  • Knowledge of mathematical operations involving negative numbers.
  • Ability to manipulate and solve inequalities correctly.
NEXT STEPS
  • Study the properties of logarithms, focusing on their behavior with negative values.
  • Learn about inequality manipulation techniques in algebra.
  • Explore real-world applications of logarithmic inequalities in various fields.
  • Practice solving inequalities that involve logarithmic expressions and negative coefficients.
USEFUL FOR

Students, educators, and anyone interested in mastering algebraic concepts, particularly those dealing with logarithmic functions and inequalities.

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What is error in the picture?
 

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The error arises because:

$$\log_{10}\left(\frac{1}{2}\right)<0$$

When we divide an inequality by a negative value, we need to reverse the direction of the inequality, so that we get:

$$2<3$$
 

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