Simplify: 9log(base27)x - 4log(base9)x
using no calculators.
Thanks in advance.
Define N such that
27^N = (9*3)^N = 3^N * 9^N = x
That means
N = log(base27) x, and
x * 3^-N = 9^N
N = log(base9)[x * 3^-N]
= log(base9)x + log(base9)(3^-N)
= log(base9)x - N log(base9)3
= log(base9)x - N/2
Therefore log(base9)x = 3N/2
and
9log(base27)x - 4log(base9)x
= 9 N - 6N = 3N
= 3 log(base27)x
The numerical answer depends upon the value of x.