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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?).

Problem Solving

If \( 3^n = 5 \) and \( 4^m = 8 \), what is the value of \[\large 9^{n+m} ? \]

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.

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 \(p+q\)?

What is the sum of all possible integer values of \(x\) such that

\[ \large (x^2-55x+755)^{(x^2-62x+936 )} =1 ? \]

Solve for \(x \):

\[ 3 = 4^x + 5^x + 6^x \]

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