HOW TO EXPAND LOGARITHMS WITH SQUARE ROOTS
Properties of logarithms :
Write the following expressions in terms of logs of x, y and z.
Problem 1 :
logx3y2z4
Solution:
=logx3y2z4=log10x3y2z4=log10x12y213z4=log10 x12+log10y213-log10 z4=12 log10 x+13×2log10 y-4 log10 z=12 log10 x+23 log10 y-4 log10 z
Problem 2 :
log x√z
Solution:
=log xz=log10 x×z=log10 x×z12=z12 log10 x
Problem 3 :
log 3x3yz
Solution:
=log3x3yz=log103x3yz=log10 x13-log10(yz)13=13 log10 x-13 log10(yz)=13log10 x-13(log10 y+log10 z)=13log10 x-13log10 y-13log10 z
Problem 4 :
log4x3y2z4
Solution:
=log4x3y2z4=log10 4x3y2z4=log10x3y2z414=log10x314y214z414=log10x314+log10y214-log10z414=14×log10 x3+14 log10 y2-14 log10 z4=14 ×3 log10 x+14×2 log10 y - 14×4 log10 z=34 log10 x+12 log10 y -log10 z
Problem 5 :
log xxz
Solution:
=log xxz=log10 xx12z=log10 xx12z12=log10 x×x1212z12=log10 x×x14z12=log10 x×x14z12=x14×log10 xz12
Problem 6 :
logxy2z8
Solution:
=logxy2z8=log10xy2z8=log10xy2z812=12×log10 xy2z8=12log10 x+log10 y2-log10 z8=12×log10 x+12×log10 y2-12log10 z8=12×log10 x+12×2 log10 y-12×8 log10 z=12×log10 x+ log10 y-4 log10 z