Number of real solutions

Equation: $$\sin\!\left(e^x\right) \;=\; 5^x + 5^{-x}$$

Reasoning

• For all real \(x\), \(\sin(e^x)\in[-1,1]\).
• By AM–GM, \(5^x + 5^{-x} \ge 2\) (with equality only when \(5^x=5^{-x}\Rightarrow x=0\)).
• At \(x=0\): LHS \(=\sin(1)\approx 0.84\), RHS \(=2\) ⇒ not equal.
• Hence RHS is always \(\ge 2\) while LHS is always \(\le 1\) → they can never match.

✅ Conclusion

No real solution (number of real solutions = 0).