find log 48 to the base 24 in terms of A if log 36 to the base 12 is A

Given:- log1236=ATo find:- log2448 First, we will simplify the   log2448 , log2448 = log2424×2                  =  log2424 +  log242      since logxy=logx+logy                  =  1 +  log242                   since logxx=1                  =  1 +  1log224                since logax=1logxa                  =  1 +  1log28×3                  =  1 +  1log28+log23                  =  1 +  1log223+log23                  =  1 +  13log22+log23             as logxn=nlogx                  =  1 +  13+log23       ...eqn1                   as log22=1We have given that, log1236=A log1262=A 2log126=A log126=A2 1log612=A2 1log66×2=A2 1log66+log62=A2 11+log62=A211+1log26=A211+1log22×3=A211+1log22+log23=A211+11+log23=A211+log23+11+log23=A21+log232+log23=A22+2log23 =2A+Alog232log23-Alog23=2A-22-Alog23=2A-2log23=2A-22-APutting the value of log2 in eqn(1)1 +  13+log23=1 +  13+2A-22-A=1 +  132-A+2A-22-A=1 +  2-A6-3A+2A-2=  6-3A+2A-2+2-A6-3A+2A-2=6-2A4-A=23-A4-ASo, value of log2448 is 23-A4-A

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