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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?).

\[\large 2^x=\dfrac{1}{2^y}, \quad\quad\quad x + y = \ ? \]

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\[\large 2^{x} = 3^{y} = 12^{z} \]

The equation above is fulfilled for non zero values of \(x,y,z\), find the value of \(\frac { z(x+2y) }{ xy }\).

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\[\large{ \begin{cases}

ab = a^b \\\\
\frac{a}{b} = a ^ {3b} \\ \end{cases}} \]

\( a\) and \(b\) are real numbers, such that \( a > 1 \) and \( b \neq 0 \), satisfying the above system.

Find \( b ^ { -a}. \)

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\[\large 5^{2x}+5^5=5^{x+3}+5^{x+2}\]

Find the sum of the solutions of \(x\) that satisfy the equation above.

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Given that \(x,y,z\) are positive real numbers that satisfy the equations: \(x = y^z, y=z^x, z = x^y \), find the value of the expression above.

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