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Exponential Functions
From compound interest to bubonic plague, things that grow or spread really fast are often modeled by exponential functions. Learn about these powerful functions (pun intended?).
Excel in math and science
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If \( 3^n = 5 \) and \( 4^m = 8 \), what is the value of \[\large 9^{n+m} ? \]
Excel in math and science
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by Brilliant Staff
Let \[ f(x) = \left( \dfrac12 \right)^{x-5} - 64, \] and suppose that the graphs of \( y = |f(x)| \) and \( y=k \) meet in the first quadrant.
Find the number of all the positive integers \( k \) that satisfy the above condition.
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by Brilliant Staff
For all positive integers \(n\), let \(\alpha_n\) and \(\beta_n\) be the roots of the quadratic equation \[nx^2-x+n(n+1)=0.\] If \[\left(\frac{1}{\alpha_1}+\frac{1}{\beta_1}\right)+\left(\frac{1}{\alpha_2}+\frac{1}{\beta_2}\right)+ \cdots +\left(\frac{1}{\alpha_{7}}+\frac{1}{\beta_{7}}\right)=\frac{p}{q},\] where \(p\) and \(q\) are coprime positive integers, what is \(p+q\)?
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by Brilliant Staff
What is the sum of all possible integer values of \(x\) such that
\[ \large (x^2-55x+755)^{(x^2-62x+936 )} =1 ? \]
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Solve for \(x \):
\[ 3 = 4^x + 5^x + 6^x \]
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by Brilliant Staff