- #1
Why?
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Homework Statement
Find all numbers x for which:
2x<8
Homework Equations
The Attempt at a Solution
I really haven't been able to figure this one out.
Inequality Problem from Spivak's Calculus: Chapter 1, Problem 4, Subproblem XI
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SUMMARY
The discussion centers on solving the inequality \(2^x < 8\) from Spivak's Calculus, specifically Chapter 1, Problem 4, Subproblem XI. Participants clarify that since \(8\) can be expressed as \(2^3\), the inequality simplifies to \(x < 3\). The increasing nature of the logarithmic function \(\log_2(t)\) is emphasized, confirming that applying it maintains the inequality. The conclusion is that all real numbers \(x\) less than \(3\) satisfy the original inequality.
PREREQUISITES- Understanding of exponential functions, specifically \(2^x\)
- Knowledge of logarithmic functions, particularly \(\log_2(t)\)
- Familiarity with inequalities and their properties
- Basic concepts from calculus as presented in Spivak's textbook
- Study the properties of exponential functions and their graphs
- Learn about logarithmic functions and their applications in inequalities
- Explore the concept of monotonic functions and their implications
- Review additional problems from Spivak's Calculus to reinforce understanding
Students of calculus, particularly those studying inequalities and exponential functions, as well as educators seeking to clarify these concepts in a classroom setting.
Find all numbers x for which:
2x<8
I really haven't been able to figure this one out.
- #2
Axiom17
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Do you have any calculations to show?
I assume it means integer values? If so surely isn't it just:
x=-\infty, ..., 0, \pm 1, \pm 2, \pm 3
Unless I'm missing the point somewhere? It seems a bit simple though.
I assume it means integer values? If so surely isn't it just:
x=-\infty, ..., 0, \pm 1, \pm 2, \pm 3
Unless I'm missing the point somewhere? It seems a bit simple though.
- #3
hunt_mat
Homework Helper
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Note that 8=2^{3}, 2^{x} in monotonically increasing. So the question is, what values of x satisfy
<br /> 2^{x}<2^{3}<br />
can you say what values satisfy this equation?
<br /> 2^{x}<2^{3}<br />
can you say what values satisfy this equation?
- #4
tmccullough
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Stated in a different way, \log_2(t) is an increasing function. Inequalities remain true if you apply an increasing function.
- #5
Why?
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Sorry I wasn't very clear.
Just thinking it through I know that 23 is 8, so x<3.
However my difficulty was in proving it, using the mathematical context that Spivak uses.
hunt_mat's example makes a lot of sense to me and fulfills that need to explain it more concretely.
Thanks for your help!
Just thinking it through I know that 23 is 8, so x<3.
However my difficulty was in proving it, using the mathematical context that Spivak uses.
hunt_mat's example makes a lot of sense to me and fulfills that need to explain it more concretely.
Thanks for your help!
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