Basic Maths and Logarithm
Logarithmic Equations
Premium Question
Grade 11
Question:
The number of solutions of $x^{(\log_{10} x)^2 - 5 \log_{10} x + 6} = 1$, $x > 0$, is:
(1) $1$
(2) $2$
(3) $3$
(4) $4$
Step-by-Step Solution
Key Concept: $a^b = 1$ when (i) $b = 0$: $(\log x)^2 - 5 \log x + 6 = 0 \Rightarrow \log x = 2$ or $3 \Rightarrow x = 100$ or $1000$. (ii) $a = 1$: $x = 1$ ($\log x = 0$, check: $0^6 = 1$? Yes, $1^0$ type but $x = 1$ gives $1^{\text{anything}} = 1\checkmark$). Total: 3 solutions $\{1, 100, 1000\}$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)