How do you solve #logₐx^10-2logₐ(x^3/4)=4logₐ(2x)#?

1 Answer
iceman
Nov 27, 2015

The equation is an identity, and therefore it has an infinite number of solutions.

Explanation:

#log_a(x^10)-2log_a(x^3/4)=4log_a(2x)#
let's work the left side#(LS)# only and simplify it:
#LS=log_a(x^10)-log_a(x^6/16)#
#LS=log_a{(x^10)/[(x^6)/(16)]}#
#LS=log_a(16x^4)#
#LS=log_a[(2x)^4]#
#LS=4log_a(2x)#
Now:
#RS=4log_a(2x)#
#:.LS=RS#
Within the basic laws and valid domain of logarithms, i.e: #a!=1 and x>0#, the original equation is an identity, and therefore it has an infinite number of solutions.

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