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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?).

# Exponential Functions - 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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