If (2.3)x=(0.23)y=1000 find 1x−1y
Given (2.3)x=(0.23)y=1000
Consider (2.3)x=1000
taking log2.3 on both sides of above equation
log2.31000=x
1log10002.3=x (Since, logmn=1lognm)
1x=log10002.3 ......(i)
Similar way Consider (0.23)y=1000
log0.231000=y
1log10000.23=y (Since, logmn=1lognm)
1y=log10000.23.....(ii)
Consider 1x−1y=log10002.3−log10000.23 (from equation (i) and (ii))
=log1032.30.23
=13log1010 (Since, logamb=1mlogab)
∴1x−1y=13 (Since, logaa=1)