SAT Question 769b612d: Nonlinear Functions answer and explanation with Desmos solution

Choice B is correct. It’s given that $g\left(x\right)=10^{5x-8}$. Since $10,000$ can be written as $10^4$, substituting $10^4$ for $g(x)$ in this equation yields $10^4=10^{5x-8}$. Because the bases on the left-hand side and right-hand side of this equation are equal, it follows that the exponents must be equal. Therefore, $4=5x-8$. Adding $8$ to both sides of this equation yields $12=5x$. Dividing both sides of this equation by $5$ yields $\frac{12}{5}=x$. Therefore, the value of $x$ when $g(x)$ is equal to $10,000$ is $\frac{12}{5}$.