\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)

Logarithms

The logarithm function is the inverse of exponentiation, just as division is the inverse of multiplication. For example, $\log_{10} 100$, which we read as "the logarithm base 10 of 100", is 2, because 100 is 10 raised to the second power. We can have logarithms of any base: for example $\log_3 81 = 4$ because $81=3^4$, $\log_5 0.2 = -1$ because $0.2=5^{-1}$, and so forth. We can even have non-integer bases; in fact the most commonly used logarithm in mathematics is the "natural log", which has $e=2.718281828\dots$ as its base. (It is so common that instead of $\log_e$ it gets its own symbol, $\ln$.)

In this book we will be most interested in the log-base-10, because it plays well with scientific notation. For instance, since $\log_{10}10^4 = 4$, then $\log_{10}(2\ten4)$ must be a little bigger than 4: in fact it is 4.3. In the old days, mathematicians had books of tables giving the logarithm for various numbers. Today, of course, we have calculators and computers. Your calculator probably has two log buttons on it, one for $\log_{10}$ and one for the natural log $\ln$. Sometimes one of the buttons will simply be labelled "log", and you'll need to deduce which one it is by looking for the other one. If you're ever not sure, try calculating "log 10". If it returns "1", then that's log-base-10.

Because logarithms are exponents, they have a couple of interesting properties. First (for any base), the log of a product is equal to the sum of the logs: $$\log ab = \log a + \log b$$ For instance, $\log_{10}(100)(1000) = \log_{10}100 + \log_{10}1000 = 2+3 = 5$, which is good because $(100)(1000)=10^5$. If you have a number in scientific notation, like $4.5\ten{-4}$, then $$\log_{10}4.5\ten{-4} = \log_{10}4.5 + \log_{10}10^{-4} = \log_{10}4.5-4$$ Since the mantissa 4.5 is between 1 and 10, its log-base-10 will be somewhere between 0 and 1, and so $\log_{10}4.5\ten{-4}$ is going to be somewhere between -4 and -3. In this case, $\log_{10}4.5 = 0.653$, so $\log_{10}4.5\ten{-4} = 0.653-4 = -3.35.

Another property that logs have is $$\log a^b = b\log a$$ One example is $\log_{10}10^6 = 6\log_{10}10 = 6(1) = 6$.