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 Functions
Logarithmic scales are used when the range of possible values is very wide, such as the intensity of an earthquake or the acidity of a liquid, in order to avoid the use of large numbers.
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
Let \(w, x, y, z \) be positive reals greater than 1, and
\[ \log_x w = 24, \log_y w = 40, \log_{xyz} w = 12 .\]
Evaluate \( \log_z w \).
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 only real solution of the equation \[\ln x + \ln (x^2-34)=\ln 72\] can be uniquely written in the form \(a+\sqrt{b}\) for some integers \(a\) and \(b\). Find \(a+b\).
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.
Given that \(\log_{10}2=0.3010\), \(\log_{10}3=0.4771\) and \(\log_{10}7=0.8451\), let \(a\) be the number of digits in \(3^{62}\). If \(b\) is the first digit of \(3^{62}\), what is \(a+b\)?
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.
Let \(N\) be the number of times that the first digit of \(2^n\) is \(1,\) with \(n \) being an integer such that \(0\leq n \leq 10^{10}\). What are the last three digits of \(N?\)
You may use the fact that
\[ 0.3010299956639811952 < \log_{10} 2 < 0.3010299956639811953 \]
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.
Evaluate \[\lfloor \log_{8} 1 \rfloor + \lfloor \log_{8} 2 \rfloor + \lfloor \log_{8} 3 \rfloor + \cdots + \lfloor \log_{8} 154 \rfloor.\]
Details and assumptions
Greatest Integer Function / Floor Function: The function \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer smaller than or equal to \(x\). For example, \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -5 \rfloor = -5\).