For [Affymetrix] microarray, the broadly accepted method of normalisation is known as Robust Multiarray Average (RMA):

1. background correction
2. quantile normalisation
3. probe summarisation (i.e. across transcripts)
4. log (base 2) transformation

Extra notes:

• An alternative to this which also adjusts for the GC content, and how it affects probe-binding affinities, is called GC-RMA.
• Other types of normalisation (step 2) exist, namely: Qspline; LOESS; VSN (variance stabilising normalisation); et cetera
• Step 3 is usually a 'median polish'
• There are intricate differences in each step based on different microarray platforms

Log transformation is not performed prior to normalisation.

For more, read the really great review by Professor Quackenbush: Microarray data normalization and transformation.

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Current RNA-seq normalisaton methods / expression measures differ quite a bit from each other. We have:

• FPKM
• RPKM
• FPKM-UQ
• RSEM
• TPM
• CPM
• TMM
• Median normalisation (DESeq2)

NB (added November 6th, 2019) - some of these are not considered normalisation procedures, per se, and are instead referred to as count measures / abundance measures / expression units that are produce from otherwise un-named normalisation procedures, e.g., FPKM

A log transformation is not typically involved in the normalisation process for RNA-seq. Statistical comparisons are performed on the normalised, unlogged counts, which generally do not follow a binomial distribution. RNA-seq count data, in fact, follows a negative binomial distribution, akin to a Poisson. However, one can later log the normalised counts, e.g. for plotting functions, in order to bring them to a binomial distribution. DESeq2, for example, implements a regularised log transformation.

For more, read A comprehensive evaluation of normalization methods for Illumina high-throughput RNA sequencing data analysis

Kevin