Question

If $lo{g}_{x}a,a^{\frac{x}{2}}$ and $lo{g}_{b}x$ are in G.P. then write the value of x.

Answer 1

$lo{g}_{x}a,a^{\frac{x}{2}},x$ are in G.P.

${\left[a^{\frac{n}{2}}\right]}^2=lo{g}_{x}a.lo{g}_{b}x$

$=\frac{loga}{logx}.\frac{logx}{logb}$

$a^x=\frac{loga}{logb}[Usinglo{g}_{y}x=\frac{logx}{logy}]$

taking log on both sides.

${\log }_{a}x=log\left(\frac{loga}{logb}\right)$

$xloga=log\left(\frac{loga}{logb}\right)$

$x=\frac{log\left(lo{g}_{b}a\right)}{loga}$

$x=lo{g}_a\left(lo{g}_{b}a\right)$