Question

If logxbase2x=a
log2xbase3x=b
log3xbase4x=c
show that abc+1=2bc??

Answer 1

remember that we can switch bases of logs using the property that
log_q p = log p /log q , where p,q > 0
and the logp/loq has the same base. So I will choose base 10, that way I don't have to write the base

then
abc = (log_2x x)(log_3x 2x)(log_4x 3x)

abc = (log x/log 2x)(log 2x/log 3x)(log 3x/log 4x)

abc = log x/log 4x
we want abc + 1 ,so

abc + 1 = (log x/log 4x) + 1
= log x/log 4x + log 4x/log 4x
= (log x + log 4x)/log 4x
= log (4x^2) / log 4x

let's work on the 2bc
2bc = 2(log_3x 2x)(log_4x 3x)
= 2(log 2x/log 3x)((log 3x/log 4x)
= 2 log 2x /log 4x
= log (2x)^2 / log 4x
= log 4x^2 / log 4x , which is abc+1

yeahhh!!