How much time it should take depends on whether or not you have access
to a calculator (even one without the exponentiation function). With a
logarithm, you can use log (base 10) to obtain
log(13^18) = 18 log 13 = 18*1.114 = 20.05
So there are twentyone digits in the number.
Now suppose you have no calculator at all (other than your brain), and
want to achieve this result in under one minute. If you know your
squares and square roots, and the rules for logarithms and exponents,
you can estimate it as follows:

We need the number log(13), to approximately two or three
digits accuracy, since we will multiply it by 18 as in the
formula above to obtain the number of digits in 13^18,
which is somewhere around twenty digits or so (it must be
more than 10^18, which has 19 digits).
13^2 = 169, which is approximately 170.
170^2 = 28900, which is around 30000,
30000^2 = 9 x 10^8 = 10^9,
finally we arrive at a number whose logarithm is easy
to evaluate, namely log(10^9) = 9.
log(13) = (1/8) log(10^9) = 9/8 = 1.125, approximately
The actual value of log(13) is 1.1139, so our estimate is
in fact good to two and almost three digits.

The rest is easy: we multiply 18 by 1.125 to obtain
18 x 9/8 = 9 x 9/4 = 81/4 = 20.25
So we conclude that there are twentyone digits.
Remark: the logarithm is very forgiving  it takes numbers that are
off by factors and converts those factors to offsets. So we may have
reasonable confidence in our estimate. Although the errors are hard to
track without a calculator, note that we rounded up twice (169 to 170
and 28900 to 30000), so that we may expect that the method above gave
a slight overestimate of log(13).
I was talking to a friend about this very interesting problem, and
we developed an even quicker route to the answer:

We estimate 13 as approximately equal to 14.142… = sqrt(2)*10.

13^18 is approximately [sqrt(2)*10]^18
= 2^9 * 10^18
= 512 * 10^18 (easy without a calculator if you remember
your powers of 2)
= 5.12 x 10^20
which has twentyone digits. Now, we can even estimate the error made
in step 1:
14.14/13 is approximately 1.1, so it is a 10% overestimate.
This overestimate will propagate when we raise it to the 18th power
(the next set of equations are approximations):
(1.1)^18 = 1.21^9 = 1.44^4 x 1.21
= sqrt(2)^4 * 1.21
= 4*1.21
= 5
This says that we have overestimated the final result by a factor of
5, which puts the actual value at approximately
13^18 = 1 x 10^20 (probably good to within 10% or so)
The actual value is 13^18 = 1.125 x 10^20.
COPIED FROM : http://mathforum.org/dr.math/ (i dont follow plagiarism)