Question
Prove that:
$a^{\log _c b}=b^{\log _c a}$
$a^{\log _c b}=b^{\log _c a}$
✓
Answer
$a^{\text {logec }^b}=b^{\text {hoge }}$
$\text { L.H.S. }=a^{\log _0 b}$
$=\left(\mathrm{e}^{\log _2 x}\right)^{\log _c b} \quad \ldots\left[x=\mathrm{e}^{\log x}\right]$
$=\left(\mathrm{e}^{\log e}\right)^{\frac{\log b}{\log c}}=\left(e^{\log b}\right)^{\frac{\log a}{\log c}}$
$=\left(\mathrm{e}^{\log _{\mathrm{B}}}\right)^{\log _{\mathrm{g}} \mathrm{s}}=\mathrm{b}^{\text {beest }}=\text { R.H.S. }$
$\text { L.H.S. }=a^{\log _0 b}$
$=\left(\mathrm{e}^{\log _2 x}\right)^{\log _c b} \quad \ldots\left[x=\mathrm{e}^{\log x}\right]$
$=\left(\mathrm{e}^{\log e}\right)^{\frac{\log b}{\log c}}=\left(e^{\log b}\right)^{\frac{\log a}{\log c}}$
$=\left(\mathrm{e}^{\log _{\mathrm{B}}}\right)^{\log _{\mathrm{g}} \mathrm{s}}=\mathrm{b}^{\text {beest }}=\text { R.H.S. }$
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