Functions
GIF and fractional part log equations
MJAT_TS2_P1
Grade 12
Question:
The number of integral values of $x$ for which $(\log_{\{x\}}[x])^2 - 3\log_{\{x\}}[x] + 2 \leq 0$ is $\lambda_1$, and the number of integral values of $x$ for which $(\log_{[x]}\{x\})^2 - 3\log_{[x]}\{x\} + 2 = 0$ is $\lambda_2$. Find $\lambda_1+\lambda_2$. ($[\cdot]$ denotes GIF, $\{\cdot\}$ denotes fractional part.)
Step-by-Step Solution
Key Concept: For $\lambda_1$: $x\in\mathbb{Z}\Rightarrow\{x\}=0$ and $\log_{\{x\}}[x]$ is undefined. So $\lambda_1=0$. For $\lambda_2$: solve $t^2-3t+2=0$ where $t=\log_{[x]}\{x\}$, giving $t=1$ or $t=2$. $t=1$: $\{x\}=[x]$ (impossible since $\{x\}<1\leq[x]$ for $x\geq 1$). $t=2$: $\{x\}=[x]^2$ (impossible for integers). So $\lambda_2=0$.
$\lambda_1=0$ (undefined for integers), $\lambda_2=0$ (no integer solutions). $\lambda_1+\lambda_2=\mathbf{0}$.
Correct Answer: 0