How does a mobile phone send signals?
There is no "power issue."  In the simplest case, base station (in the cell tower) transmits to your phone with about the same power level that your phone transmits.  

The base station also transmits to other phones at the same time so the total transmitted power is higher.  It "talks" to each phone no "louder" than necessary, though.

You're thinking of the base station like a public address speaker - big and loud to get the message out as far as possible, and you wonder how the little phone can match that for distance.  In reality, it's more like an intercom - no louder than the "speaking" voice of those who can hear it.  The base station and the phone "hold a conversation" at the volume of the "speaking voice" of the phone.



The base stations use large antennas because it helps with transmitting and receiving.  The base station uses the same antenna to transmit as well as to receive.  A big, efficient antenna helps the base station to "hear" your phone better.  At the same time, the big, efficient antenna makes the base station's "voice" louder so your phone can "hear" it better.  

The actual power transmitted from the base station towards your phone is about the same as the power your phone transmits towards the base station.

Cell phone towers often have more than one antenna, so it looks like there's an even bigger difference in antenna size than there really is.

The multiple antennas on a cell phone tower are arranged to cover different areas, and may have multiple base stations behind them to handle the traffic.

Each phone channel can only handle a certain number of users at once, and there is only a certain number of channels available.  Since you can't change those two things, you use directional antennas and divide the area around the tower into sectors - each sector is then covered by an antenna and base station.  You can get more users on the system that way.



The big antennas are both an ear trumpet and a megaphone at the same time.