Question

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

Answer 1

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.