Math Problem Statement
Please solve this
Solution
Let's solve each problem one by one from the image provided. I'll focus on each inequality and logarithmic equation.
Problem 492
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Rewrite as :
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Multiply both sides by 2:
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Rewrite as :
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Since the logarithmic function is increasing, we can remove the logs:
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Rearrange this inequality:
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Factor:
Since is true for all , the solution under the constraint is:
Problem 493
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Let , then the equation becomes:
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Rearrange this as a quadratic inequality:
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Solve the quadratic inequality for using the quadratic formula:
Solving this will give values of , which you can then substitute back for .
Problem 494
This is a complex inequality involving both and . We can let and rewrite it as: Then solve for and back-substitute for .
Problem 495
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Simplify using logarithmic properties:
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Factor out the 2:
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Simplify further:
Then solve for from this inequality.
Problem 496
Since is a decreasing function, we can reverse the inequality: Then solve for .
Let me know if you'd like detailed steps for each solution or if there are specific parts you need more clarification on!
Questions for Further Exploration
- What happens if you have an inequality with mixed bases in logarithms?
- How do properties of logarithms change when the base is between 0 and 1?
- How do you handle equations where logarithmic expressions are squared?
- What additional techniques help with inequalities involving fractional logarithmic expressions?
- Can logarithmic inequalities have solutions that need domain restrictions?
Tip
For inequalities with logarithmic terms, consider converting all terms to the same base or variable if possible, as this can greatly simplify the problem.
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Math Problem Analysis
Mathematical Concepts
Logarithmic Inequalities
Logarithmic Properties
Algebraic Manipulations
Quadratic Inequalities
Formulas
log_a (x^m) = m * log_a(x)
log_a (xy) = log_a(x) + log_a(y)
Change of base formula: log_b(x) = log_c(x) / log_c(b)
Theorems
Properties of Logarithms
Rules of Inequalities (e.g., reversing inequality when multiplying by a negative)
Suitable Grade Level
Grades 10-12