HOW TO ESTIMATE COMMON AND NATURAL LOGARITHMS IN YOUR HEAD
The fact that we humans evolved to have ten fingers (which naturally led our species to use a base-10 number
system) causes three wonderful coincidences that, together with a thorough understanding of the properties of
logarithms, make it relatively easy to estimate the values of certain common (base-10) and natural (base-e)
logarithms without the benefit of an electronic calculator. These somewhat unlikely-looking coincidences are:
210  1024  1000  103
35  243  250
e23  9744803446.249  1010
where, in the last equation, e  2.718281828459 is the irrational base for the natural logarithm. Although each
of the above approximations is off by nearly three percent, we can still obtain remarkably accurate results by
using them as if they were equalities. In the discussion that follows, per common usage, “log” will be an
abbreviation for the common logarithm, and “ln” will be an abbreviation for the natural logarithm. Let us start by
assuming the first equation above to be exact:
210  103  log 210  log103  10log 2  3log10  10log 2  3  log 2  0.30
A calculator tells us this is an excellent approximation; the actual value of log 2 is 0.30103…. This single result
allows us to determine the common logarithm of any power of 2, with an accuracy of better than one percent:
log 4  log 22  2log 2  0.60; log8  log 23  3log 2  0.90; log  161   log 24  4log 2  1.20 , etc.
With just a slight additional effort, this result also allows us to find the common logarithm of 5 (and, by extension,
of any power of 5).
log5  log 102  log10  log 2  1.00  0.30  0.70 .
By using the product and/or quotient rules for logarithms, we can calculate the common logarithm of any number
whose prime factors consist exclusively of 2s and/or 5s by simply adding or subtracting the appropriate
logarithms. For example,
log 250  log(2  53 )  log 2  3log5  0.30  3(0.70)  2.40; or (done another way),
log 250  log  1000
 log
4 
   3log10  2log 2  3  2(0.30)  2.40.
103
22
Considering the integers between 1 and 100, we can now estimate the logs of any number in the following list:
{1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100}. In addition, we can find the logarithm of any fraction whose
numerator and denominator consist of these numbers (or their powers). But this is only the beginning!
Let us examine the second coincidence listed at the beginning of this article. Since we have just calculated the
approximate logarithm of 250, we can write (note the carelessness with which we switch between  and =):
35  250  log35  log 250  5log3  2.40  log3  0.48
Again, this gives excellent agreement with the “true” value of log3 , which a calculator assures us is 0.47712….
(Note that the accuracy here is still better than one percent.) With this enhancement, we can now calculate the
logarithm of any number whose prime factors include 2, 3, and/or 5. From the first 100 positive integers, the
subset whose logs we can now estimate with great accuracy is {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25,
27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100}, or more than a third!
We can “guesstimate” some of the remaining logarithms by using a technique called “linear interpolation”. Over
short segments of the curve representing the logarithm function, this curve is nearly linear. So if a number lies
halfway between two other closely-spaced numbers whose logs are known, the logarithm of that number will lie
roughly halfway between the known logs. A table of the common logs of the first twelve positive integers, as
estimated using the previous techniques, allows us to guess at the missing ones (represented by question marks):
x
1
2
3
4
5
6
7
8
9
10
11
12
log x
0
0.30
0.48
0.60
0.70
0.78
?
0.90
0.96
1.00
?
1.08
VC DEPARTMENT OF MATHEMATICS
REVISED SPRING 2010
Note that the spacing of logs for smaller consecutive numbers is greater than for larger consecutive numbers (that
is, the spacing between the values of log 2 and log 3 is more than the spacing between the values of log 3 and
log 4). Since 7 is halfway between 6 and 8, we estimate that log 7 is slightly more than halfway between log 6
(0.78) and log 8 (0.90). This would place our estimate at 0.84 (exactly halfway) or 0.85 (a little more than
halfway). The actual value is 0.84510…, so 0.85 is better. Using similar reasoning, we estimate log 11 as 1.04 or
1.05 (actual value: 1.04139…, so 1.04 is better in this case). Filling in our table with these values gives
x
1
2
3
4
5
6
7
8
9
10
11
12
log x
0
0.30
0.48
0.60
0.70
0.78
0.85
0.90
0.96
1.00
1.04
1.08
So now, the only numbers whose logs we cannot easily estimate are those containing a prime factor of 13 or
greater.
What about natural logarithms? If we know the common logarithm, then all we need to do is apply the change-ofbase formula. For this, we need to know the value of either ln 10 or log e. For our purposes, ln 10 is more useful.
We can estimate this value by using the third “coincidence” equation from the beginning of this article:
e23  1010  ln e23  ln1010  23ln e  10ln10  10ln10  23  ln10  2.3
We continue to maintain an accuracy of better than one percent, since ln 10 (with a calculator) is 2.302585…. The
change-of-base formula tells us that
log  some number  
ln  some number 
ln10
or, equivalently,
ln  some number    ln10  log  some number   ln some number   2.30  log  some number  .
If we know the common logarithm of any number, this formula tells us that all we have to do to obtain the natural
logarithm of the same number is multiply by 2.30! This may seem intimidating at first, but it is actually no more
difficult than calculating a 15% tip at a restaurant. (Sorry if you think that 15% is a cheap tip, but that’s just the
way the math works out.) Note that if we let x represent the cost of a meal (before the tip is added), then 1.15x is
the total cost after the tip is added in. If we then double this amount (imagine that you took your sweetie out to
dinner, and you both ordered the same thing), we obtain the desired result (2.30x). So treat the common logarithm
as the pre-tip price of one meal; the desired natural logarithm becomes the post-tip cost of two meals. Note that it
doesn’t matter whether you calculate the tip first, then double, or vice-versa.
As an example, let’s find ln 30. First, find the common logarithm of 30, using our short table from above:
log30  log  2  3  5  log 2  log3  log5  0.30  0.48  0.70  1.48 .
To find ln 30, imagine that you ordered a meal whose price (before tip) was $1.48. Since you’re a big spender,
you splurged for your date too, bringing the total to $2.96. Rounding this to $3.00 to make the math easier, the tip
will be about $0.45 (that is, fifteen cents tip on each of the three dollars you spent). Add this to the $2.96 (use the
original bill amount for this step, not the rounded amount), and you get a total tab of $3.41, or just 3.41. This
compares with the calculator’s result of 3.401197…. Even though we are off by a penny, our result is still
accurate to much better than one percent.
To summarize, you can accomplish a lot by memorizing just three numbers: log 2  0.30 , log3  0.48 , and
ln10  2.30 . Of course, you will need some practice in order to make yourself comfortable with the various
logarithm rules before you can estimate logs in your head reliably!
VC DEPARTMENT OF MATHEMATICS
REVISED SPRING 2010