1.9 Change of Base

We started our discussion of logarithms by looking at the number 2 raised to various powers, \(p\). We soon saw that using 10 as our base number was more congenial for most general multiplication and division problems. Logarithms using 10 as their base were called common logarithms. But clearly the rules for using logarithms are the same no matter what base we choose.

So now suppose one has knowledge of the logarithm of a number for a particular base, say \(b\). We can find the logarithm of the same number relative to a different base, say \(a\), via the following argument. If

\[ x = a^p \]

then taking the logarithm using base \(b\) on both sides,

\[ \log_b x = \log_b a^p =p\times \log_b a \]

using the rule we discussed earlier. But note that \(p\) is the logarithm of \(x\) using base \(a\), and so

\[ \log_b x = \log_a x \times \log_b a. \]

Thus, a switch from base \(b\) to base \(a\) is performed by

\[ \log_a x = \log_b x/\log_b a. \]

More than just an interesting tidbit, this is actually an important result. It tells us that we do not need hundreds of tables of logarithm values for the vast number of different bases that might be of interest in different situations. We actually only need one table of values. If we have the logarithms tabulated for one base, we can easily compute values for any other base, as needed. From our historical path, and through a bit of evolution, we have become a “Base 10” civilization, and so logarithms using that base are chosen as the common logartihms. The development of the calculus and of functional analysis in mathematics shows that using the constant e = 2.718… is a natural choice for describing exponential growth and decay; logarithms with e as their base are called natural logarithms. (See The Natural Logarithm.) In mathematical parlance, the common logarithm of \(x\) is denoted by \(\log x\) while the natural logarithm is denoted by \(\ln x\). If yet a different base \(b\) is used, it is denoted by \(\log_b x\) as used earlier.

As an illustration of exchanging bases, again look at \(y=2^{3.659}\). Taking the logarithm using Base 2,

\[ \log_2 y = 3.659 \]

and since \(\log_2 y = \log y / \log 2\), then

\[ \log y = \log_2 y \times \log 2 = 3.659\times\log 2 = 3.659\times 0.30103 = 1.10147 = 0.10147 + 1. \]

The number with common logarithm 0.10147 is 1.2632, and so it must be that \(y=12.632\) as we found previously. (See Exponents and Powers.)

As another example, suppose we also needed the natural logarithm of our result \(y=12.632.\) We can do so by noting that

\[ \ln y = \log y / \log e. \]

For reference, the common logarithm of \(e\) is \(\log e\) = 0.43429, and, when needed, the natural logarithm of 10 is \(\ln 10\) = 2.30259. Hence,

\[ \ln y = 1.10147 / 0.434 = 2.536 \]

and, correspondingly, \(y = e^{2.536}\) = 12.632.