There are no blockchain transactions within an exchange.
Let me explain: (Y = you, B = someone else, C = someone else, E = exchange, assume that all people have 0 BTC, but C has 1 BTC so far)
You send 1 BTC as a blockchain transaction to the adress of E.
=> E makes a database entry "Y have 1 BTC".
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B buys 1 BTC from C
=> E makes a database entry "B has 1 BTC" and "C has 0 BTC" (no blockchain transaction).
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B sell his 1 BTC to C
=> E makes a database entry "B has 0 BTC" and "C has 1 BTC" (no blockchain transaction).
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You transfer your BTC to your own adress as a blockchain transaction
=> E makes a blockchain-transaction and an entry "Y have 0 BTC".
They don't need to make blockchain-transaction within their service. They do just edit their database which tells them who has how many BTC. And as you don't have access to their database, you can't see those edits.
Sounds a bit centralised, if you ask me ;) Thanks Alpha! – Kyle Graham – 2018-01-12T16:28:40.610
1Of course it is. It wouldn't be any less centralized if they used a "private blockchain" internally to do the same. They are custodians over your money. – Pieter Wuille – 2018-01-12T17:46:53.640