The growth rate function for a nonempty minor-closed class of matroids
M
\mathcal {M}
is the function
h
M
(
n
)
h_{\mathcal {M}}(n)
whose value at an integer
n
≥
0
n \ge 0
is defined to be the maximum number of elements in a simple matroid in
M
\mathcal {M}
of rank at most
n
n
. Geelen, Kabell, Kung and Whittle showed that, whenever
h
M
(
2
)
h_{\mathcal {M}}(2)
is finite, the function
h
M
h_{\mathcal {M}}
grows linearly, quadratically or exponentially in
n
n
(with base equal to a prime power
q
q
), up to a constant factor.
We prove that in the exponential case, there are nonnegative integers
k
k
and
d
≤
q
2
k
−
1
q
−
1
d \le \tfrac {q^{2k}-1} {q-1}
such that
h
M
(
n
)
=
q
n
+
k
−
1
q
−
1
−
q
d
h_{\mathcal {M}}(n) = \frac {q^{n+k}-1}{q-1} - qd
for all sufficiently large
n
n
, and we characterise which matroids attain the growth rate function for large
n
n
. We also show that if
M
\mathcal {M}
is specified in a certain ‘natural’ way (by intersections of classes of matroids representable over different finite fields and/or by excluding a finite set of minors), then the constants
k
k
and
d
d
, as well as the point that ‘sufficiently large’ begins to apply to
n
n
, can be determined by a finite computation.