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3 problems match “exponential equations”. Clear

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1. Real roots of an exponential-cosine equation ●●○○○ 20 ptsFind the number of real roots of the equation $e^{\cos x} - e^{-\cos x} - 4 = 0$.
2. Coin flips moving two numbers together ●●●○○ 50 ptsSteph and Jeff each start with the number 4, and Travis is flipping a coin. Every time he flips heads, Steph replaces her number $x$ with $2x-1$, and Jeff replaces his number $y$ with $y+8$. Every time he flips tails, Steph replaces her number $x$ with $\frac{x+1}{2}$, and Jeff replaces his number $y$ with $y-3$. After some (positive) number of coin flips, Steph and Jeff end up with the same number, which is below 2012. How many times was the coin flipped?
3. Exponential equation with a cube root ●●●●○ 100 ptsSolve the equation $27^{x}-7 \sqrt[3]{7 \cdot 3^{x}+6}=6$.