Basic Mathematics & Logarithm
Logarithmic Equations — System of Log Identities
nta_pyq_2023_apr
Grade 11
Question:
Let $a,b,c$ be three distinct positive real numbers such that $(2a)^{\ln a}=(bc)^{\ln b}$ and $b^{\ln 2}=a^{\ln c}$. Then $6a+5bc$ is equal to ______.
Step-by-Step Solution
Key Concept: Take $\ln$ of both equations. Let $x=\ln a,\ y=\ln b,\ z=\ln c$. First equation: $x(\ln 2+x)=y(y+z)$; second: $y\ln 2=xz$. Substitute to derive $(x^2-y^2)(y+z)=0$.
$(y^2-x^2)(y+z)=0$. $x=-y\Rightarrow \ln 2=-\ln a\Rightarrow a=\frac{1}{2}$; $y=-z\Rightarrow bc=1$. $6a+5bc=3+5=8$.
Correct Answer: 8