Prove that log base a of x divided by log base a/b of x is equal to 1+log base a of(1/b)

Where do I start?

1 answer

since
log_u(x)/log_v(x) = log_v(u) and
log_u(x) = 1/log_x(u), we have

log_a/b(a) = 1/log_a(a/b)
= log_a(a) + log_a(1/b)
= 1 + log_a(1/b)

If those two properties of logs have not yet been delivered to you, work to convince yourself that they are true. I'm sure you can find many proofs online.