Question

If log 9(18) = x+1 then log 9(24) is equal to

Answer 1

$\textbf{Given:}$

$\log_{9}18=x+1$

$\textbf{To find:}$

$\text{The value of \log_{9}24 }$

$\textbf{Solution:}$

$\text{Consider,}$

$\log_{9}18=x+1$

$\implies\,9^{x+1}=18$

$\implies\,9^{x}=2$

$\implies\,(3^2)^{x}=2$

$\implies\,3^{2x}=2$

$\implies\bf\,3=2^{\frac{1}{2x}}$

$\text{Now,}$

$\text{Let}\;\log_{9}(24)=y$

$\implies\,9^y=24$

$\implies\,(3^2)^y=24$

$\implies\,3^{2y}=3{\times}2^3$

$\implies\,3^{2y-1}=2^3$

$\implies\,(2^{\frac{1}{2x}})^{2y-1}=2^3$

$\implies\(2^{\frac{2y-1}{2x}}=2^3$

$\text{Equating powers on both sides we get}$

$\dfrac{2y-1}{2x}=3$

$\implies\,2y-1=6x$

$\implies\,2y=6x+1$

$\implies\,y=\dfrac{6x+1}{2}$

$\implies\bf\,log_{9}24=\dfrac{6x+1}{2}$

$\therefore\textbf{The value of \bf\,log_{9}24 is \bf\dfrac{6x+1}{2} }$

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