2.5 Computing Common Logarithms

For common logs, we can write any number in scientific notation and compute its logarithm by breaking it into two parts. Suppose \(x\) is written in scientific notation as \(x = a \times 10^n\) where \(1\le a < 10\) and \(n\) is an integer. Then

\[ x = a/10 \times 10^{n+1} \]

and

\[ \ln x = \ln(a/10) + (n+1)\; \ln 10 \]

from which

\[ \log x = \frac{\ln x}{\ln 10} = \frac{\ln (a/10)}{\ln 10} + n+1 . \]

Since \(a\) was defined to be between 1 and 10, then \(a/10\) is between 0.1 and 1 and we can use our power series expansion for \(\ln(x)\) in our calculations. Notice that \(\ln(a/10)\) necessarily will be negative in this instance. We are now in a position to create a function to compute common logarithms for general numbers. Here is an example computer code to do just that, re-using our previous function for computing natural logarithms:

logCom = function(x){                                # ex:   x = 2.827
   Num = strsplit(format(x, scientific=T),"e")[[1]]  # split input into two strings:  Num = c(".827","2")
   a   = as.numeric(Num[1])                          # take the 2 parts and make them numeric:  a = 0.827, 
   n   = as.integer(Num[2])                          #                                          n = 2
   lnX(a/10)/ln10 + n + 1                            # i.e., use our formula...
}

Below are a few examples, using our new function, of the calculation of various common logarithms:

log 0.718 = -0.14388	log 1.25 = 0.09691	log 2 = 0.30103
log e = 0.43429	log 10 = 1	log 42.5 = 1.62839
log 200 = 2.30103	log 1050 = 3.02119	log 3.4\(\times 10^5\) = 5.53148