if-log-a-b-log-b-c-log-c-a-show-that-a-b-c-

Question Number 25315 by ibraheem160 last updated on 08/Dec/17

i f {l o g}_{a}^{b} = {l o g}_{b}^{c} = {l o g}_{c}^{a} . s h o w t h a t a = b = c

Answered by prakash jain last updated on 08/Dec/17

\log_{a} b \log_{b} c = \log_{a} c

\Rightarrow {(\log_{c} a)}^{2} = \log_{a} c

\Rightarrow 1 = {(\log_{a} c)}^{3} \Rightarrow \log_{a} c = 1

\Rightarrow a = c

\log_{c} a = \log_{b} c = \log_{a} b = 1

\Rightarrow a = b = c

Answered by Rasheed.Sindhi last updated on 08/Dec/17

Let \log_{a} b = \log_{b} c = \log_{c} a = y

a^{y} = b, b^{y} = c, c^{y} = a

a^{y} = b, b^{y} = c \Rightarrow {(a^{y})}^{y} = c

c^{y} = a, {(a^{y})}^{y} = c \Rightarrow {{(c^{y})}^{y}}^{y} = c

c^{y^{3}} = c \Rightarrow y^{3} = 1 \Rightarrow y = 1

a^{y} = b \Rightarrow a^{1} = b \Rightarrow a = b

b^{y} = c \Rightarrow b^{1} = c \Rightarrow b = c

a = b, b = c \Rightarrow a = b = c