I am thoroughly confused on this topic. How does "common base" even make sense? If the base is grounded then the transistor is turned off, so how does the input reach the output?
Answer
The fact that the base is common to the input and output signals does not imply that the base is at the lowest voltage (assuming npn). Compare with common collector (aka emitter follower): the collector is at the highest voltage, not at the lowest!
In the common base configuration the base is held at a fixed voltage, the input is applied to the emitter, the output is taken from the collector. The input signal will be loaded heavily (input impedance is very low), output impedance is high, current amplification is ~ 1, voltage amplification is very high.
This configuration is sometimes used in HF stages.
With some hand-waving the long-tailed pair (input stage of an opamp) with one input fixed can be seen as an emitter follower + a common base.
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