Question

If $5^x+{\left(2\sqrt{3}\right)}^{2x}\ge {13}^x$, then the smallest positive integer excluded from the solution is

Answer 1

If $5^x+{\left(2\sqrt{3}\right)}^{2x}\ge {13}^x$

$5^x+{12}^x\ge {13}^x$

${25}^{x/2}+{144}^{x/2}\ge {169}^{x/2}$

At $x=2$, the condition is true $\frac{x}{2}\le 1$.

If the power is less than $1$, then $x\in (-\infty ,2]$.

So, the smallest value excluded from the solution is $3$.