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  •  If x, y, z are in G.P. and  ax=by=cz, then           Option: 1 <i

 If x, y, z are in G.P. and  ax=by=cz, then           

Option: 1

\begin{array}{l}{\log _{a} c=\log _{b} a}\end{array}


Option: 2

\begin{array}{l}\\ {\log _{b} a=\log _{c} b}\end{array}


Option: 3

\begin{array}{l} \\ {\log _{c} b=\log _{a} c} \end{array}


Option: 4

None of these


Answers (1)

best_answer

\\\text{Given x,y, z is in G.P. so } y^2=x \cdot z ... (i)\\\\ a^x=b^y=c^z=t\\ \\\text{Apply log on both side}\\\\ x \log_{e}a=y \log_{e}b=z \log_{e}c= \log_{e}t

\\ x=\frac{\log_{e}t}{\log_{e}a},y=\frac{\log_{e}t}{\log_{e}b},z=\frac{\log_{e}t}{\log_{e}c}\\\\ \text{from equation (i)}\\\\ \left (\frac{\log_{e}t}{\log_{e}b} \right )^2=\frac{\log_{e}t}{\log_{e}a} \cdot \frac{\log_{e}t}{\log_{e}c}\\\\\\ \frac{\log_{e}a}{\log_{e}b}=\frac{\log_{e}b}{\log_{e}c}\\\\\log_{b}a=\log_{c}b

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Rakesh

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