I have the equation:
$.5 = 1-(1-s^r)^b$
And want to solve for $s$.
Would the correct solution involve (1) b-roots and r-roots or (2) logarithms?
Here's approach (1):
Here's approach (2):
Approach (1) seems to be less messy algebraically, but beyond that am I missing something that makes one approach better (or more correct) than the other?
You are not missing anything. When I teach Calc I, I teach my students both of these approaches.
Compare the following four equations: $2^x=b$, $x^b=e$, $x^by^c=f(y)$, and your equation. To be solved algebraically: in the first case, you have to take log$_2$; in the second case, you have to raise to $1/b$ power; in third case, you have to divide by $y^c$; in your case, you have to take a root. And it's hard for first-time students to figure out which one to do when.
The log approach, however, is failsafe. One approach works for all four types of equations. The drawback is, log is much slower than the algebraic approach.
When you get to logarithmic differentiation, most of those problems can be solved without log as well (e.g., $x^x=e^{x\log x}$; then use chain rule to differentiate). However, logarithmic differentiation is often taught just because it's one uniform approach that works for all types of problems, including $(x+2)^2(x+1)^3(x-3)^4$.