LM394 is obsolete. What is the new standard log amp circuit?

The Analog Devices SSM2212 is a cheaper ($2.50 in quantity) alternative to the MAT12, with similar headline specs.

Cheaper matched NPN pairs include the DMMT3904W and DMMT5551 from Diodes Inc., the PMP4501, PMP4201 and BCM847 from NXP, and the NST45011M/NST65011M from ON Semi, which are roughly an order of magnitude cheaper than the SSM2212 but have a maximum offset voltage an order of magnitude worse (1-2mV).

As Roman mentions in another answer, at the time of writing (2020), Alfa RPAR make the AS394 (http://www.alfarzpp.lv/eng/sc/AS394CH.php), an LM394 substitute stocked by a number of specialist electronic music distributors.

How about the LS312 from Linear Systems? Some people in other forums claim it's better than the 5% worst-case matching described on the datasheet.

I remember some years ago using the Analog Devices MAT02 for this purpose. It seems that this is not recommended for new designs but the replacement is the MAT12.