The equation 3^x = x + 10 has a positive real solution. Between which two consecutive integers does this positive solution lie?

Define $f(x) = 3^x - x - 10$, so a solution of the equation is a zero of $f$. Test integers: $f(1) = 3 - 1 - 10 = -8$, $f(2) = 9 - 2 - 10 = -3$, and $f(3) = 27 - 3 - 10 = 14$. The function changes from negative to positive between $x = 2$ and $x = 3$, and since $f$ is continuous, a zero lies in between. That is choice B. Choice A and choice D fail because $f$ is negative at both ends of those intervals. Choice C fails because $f$ is already positive at $x = 3$ and grows rapidly afterward. A sign change brackets the root.