Logarithms
System of log equations — two solutions
MJAT_TS6_P1
Grade 12
Question:
Let $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ satisfy the system:
$\log_{10}(2xy)=4+(\log_{10}x-1)(\log_{10}y-2)$
$\log_{10}(2yz)=4+(\log_{10}y-2)(\log_{10}z-1)$
$\log_{10}(zx)=2+(\log_{10}z-1)(\log_{10}x-1)$
with $x_1>x_2$. Match List-I with List-II:
P) $y_1/x_1$; Q) $z_1\cdot x_2$; R) $z_1\cdot x_2/z_2$; S) $y_2+z_1\cdot x_2$
A) P-1; Q-2; R-4; S-3
B) P-1; Q-2; R-3; S-4
C) P-2; Q-1; R-4; S-3
D) P-2; Q-1; R-3; S-4
Step-by-Step Solution
Key Concept: Let $A=\log_{10}x-1$, $B=\log_{10}y-2$, $C=\log_{10}z-1$. The system becomes: $\log_{10}(2xy)=4+AB$, $\log_{10}(2yz)=4+BC$, $\log_{10}(zx)=2+CA$. Expand and solve for $(A,B,C)$.
P→(1), Q→(2), R→(3), S→(4). Answer: **B**.
Correct Answer: B