Waste less time on Facebook — follow Brilliant.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Logarithmic Inequalities
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Concept Quizzes
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
The sound intensity level (measured in decibels) is given by the formula: \[L_I=10\log\left(\frac{I}{I_0}\right)\text{ (dB)},\] where \(I\) denotes sound intensity and \(I_0\) is the reference sound intensity.
An individual is diagnosed as moderate hearing loss when his/her threshold of sound perception is greater than 40 dB and less than or equal to 55 dB. Given that \(I_0=1\text{ pW/m}^2,\) what is the range of the threshold of sound intensity \(x\) that a person with moderate hearing loss can perceive?
Details and Assumptions:
- The sound intensity level of a normal conversation is about 50~60 dB, and that of a quiet library is about 40 dB.
- Assume that \(\sqrt{10}\approx3.\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
\(A, B \) and \(C\) are positive integers satisfying \( A \geq 2, B \geq 4, C \geq 8 \). What is the minimum integer value of
\[ \log_A (BC) + \log_B (CA) + \log_C (AB) ? \]
Details and assumptions
The minimum integer value of a set, is the smallest integer that appears in the set. As an explicit example, the minimum integer value of the set of numbers \(x\) satisfying \( x \geq 12.3\) is 13.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If one root of the following quadratic equation \[x^2-2x\log_{9}a+4\log_{9}a=0 \] is positive and the other root is negative, what is the range of \(a?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If the range of \(x\) that satisfies the inequality \[{4}^4 {7}^4 x^3 > x^{\log_{28}x}\] is \(\alpha < x < \beta,\) what is the value of \(\alpha\beta ?\)