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How Many Digits in 13^18 ?

Date: 05/15/2003 at 01:58:33
From: Priya
Subject: Number of digits in 13 to the power of 18 (13^18))

Hi,

Find the exact number of digits in 13 raised to the 18th power.

I tried doing it using the binomial expansion method, but it became 
too cumbersome. Since this question is from a popular entrance test 
where the test taker is supposed to take only 45 secs for one 
question, I guess there should be an easier way to do it.


Date: 05/15/2003 at 15:28:54
From: Doctor Douglas
Subject: Re: Number of digits in 13 to the power of 18 (13^18))

Hi Priya,

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 twenty-one 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:

   1.  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.

   2.  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 twenty-one 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:

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

2.  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 twenty-one 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.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 


Date: 05/16/2003 at 00:44:12
From: Priya
Subject: Thank you (Number of digits in 13 to the power of 18 (13^18))

Thanks a lot for the prompt reply. Really appreciate it.

Both the methods were quite easy, though they used different 
approaches. Thanks again.
Associated Topics:
College Exponents
High School Exponents

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