From this formula by putting n = 1 we can obtain \log_e 2. Again by putting n=2 we obtain \log_e3 -\log_e2; whence \log_e3 is found, and therefore also \log_e9 is known.
Now by putting n = 9 we obtain \log_e10 - \log_e9; thus the value of \log_e10 is found to be 2.30258509 \ldots.
To convert Napierian logarithms into logarithms to base 10 we multiply by \frac{1}{\log_e10}, Which is the modulus [Art. 441] of the common system, and its value is \frac{1}{2.30258509} or .43429448\ldots; we shall denote this modulus by M.
By multiplying the last series throughout by M we obtain a formula adapted to the calculation of common logarithms.
Thus M \log_e(n +1) - M \log_e n =
that is, \log_10(n +1) - \log_10 n =
Hence if the logarithm of one of two consecutive numbers be known, the logarithm of the other may be found,
and thus a table of logarithms can be constructed.
EXAMPLES XLVI.
1. Show that
(1) (2)
2. Expand log 1 + x in ascending powers of x.
3. Prove that \log_e 2 =
4. Show that \log_10 \frac{1}{1-x}= .
5. Prove that \log \frac{1+x}{1-3x} =
6. Show that if x > 1, log x^2 -1 =
