GATE2018 ME-2: GA-5

Question

The value of the expression $\dfrac{1}{1+ \log_u \: vw} + \dfrac{1}{1+ \log_v \: wu} + \dfrac{1}{1+\log_w uv}$ is _______

• A. $-1$
• B. $0$
• C. $1$
• D. $3$

Answer 1

$\dfrac{1}{1+ \log_{u} \: vw} + \dfrac{1}{1+ \log_{v} \: wu} + \dfrac{1}{1+\log_{w} uv}$

$ = \dfrac{1}{\log_{u} \: u + \log_{u} \: vw} + \dfrac{1}{\log_{v} \: v  + \log_{v} \: wu} + \dfrac{1}{\log_{w} \: w +\log_{w} \: uv}$

$ = \dfrac{1}{\log_{u} \: uvw} + \dfrac{1}{\log_{v} \: vwu} + \dfrac{1}{\log_{w} \: wuv}$

$ = \dfrac{1}{\log_{u} \: uvw} + \dfrac{1}{\log_{v} \: uvw} + \dfrac{1}{\log_{w}\: uvw}$

$ = \log_{uvw} \: u + \log_{uvw} \: v + \log_{uvw} \: w$

$ = \log_{uvw} \: uvw $

$ = 1\qquad\because\left(\log_{a} \: a = 1 \right)$

Hence $(C)$ is Correct.

Comments

assume  u=v=w=10

we will get ans as 1

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why just u=v=w=10 only…

you can also take

u=v=w=100

u=v=w=1000

u=v=w=2

u=v=w=3

.

.

u=v=w =K

K is any number for which log is defined.