For a scalar x,

log2(x) = log(x)/log(2)

I see no reason why this wouldn't work with matrix logarithms: logm(m)/log(2).

For example, let's take the matrix from this example on Wikipedia:

issimilar = @(x,y) all( abs(x(:)-y(:)) < 1e-14 );

m = [1.25, 0.75; 0.75, 1.25];

issimilar( exp(1)^logm(m), m ) % returns true
issimilar( 2^(logm(m)/log(2)), m ) % also returns true

You can plot directly using the plot command

plot (log2(x), y)

but then your x ticks will be the logarithm rather than the actual value. You could either just change your label

xlabel('Log (base 2) of quantity X');

or you can redo the ticks manually.

xt = get(gca, 'XTick');
set (gca, 'XTickLabel', 2.^xt);

Or you can be really fancy

xticks = 10:25;
set(gca, 'XTick', xticks);
for j = 1:length(xticks)
  xtl{j} = ['2^' num2str(xticks(j))];
end
set(gca, 'XTickLabel', xtl)

which will evenly space the tick marks on the log scale, and label them according to their power of 2

I would wager for the exact same reason we use base 10 for common log, the nonfractional part tells you the length in bits (for base 10 it tells you the number of digits). Not to mention the fact that computers generally do math more native in binary. As is mentioned in the comments, all logarithms are just a constant multiple of each other, so it is really up to taste.