Elsa tries to solve the following equation, and determines there is no solution. Is she correct? Explain. log2x = log2(3x + 5) + 4

Answer:

Elsa is correct; there is no solution. To solve the equation $\log_2(x) = \log_2(3x + 5) + 4$, first combine the logarithmic terms: $\log_2(x) - \log_2(3x + 5) = 4$, which simplifies to $\log_2(\frac{x}{3x + 5}) = 4$. Converting to exponential form gives $\frac{x}{3x + 5} = 2^4 = 16$. Solving for $x$: $x = 16(3x + 5) \implies x = 48x + 80 \implies -47x = 80 \implies x = -\frac{80}{47}$. However, in the original equation, the domain for $\log_2(x)$ requires $x > 0$. Since $x = -\frac{80}{47}$ is negative, it is an extraneous solution. Because this is the only potential solution, the equation has no real solution.