Features and Properties of Logarithm

Several properties follow directly from the definition $a^{x}=N \Leftrightarrow \log_{a}N = x$. They are used constantly in simplification and should be understood, not merely memorised.

1. The logarithm of 1 to any base is zero: $\log_{a} 1 = 0$. This is because $a^{0} = 1$ for every valid base $a$. So $\log_{10}1 = 0$, $\log_{2}1 = 0$, and so on.
2. The logarithm of the base to itself is one: $\log_{a} a = 1$, because $a^{1} = a$. Thus $\log_{10}10 = 1$ and $\log_{e}e = 1$.
3. The logarithm of a power of the base equals the exponent: $\log_{a}(a^{n}) = n$. Hence $\log_{10}100 = \log_{10}(10^{2}) = 2$ and $\log_{10}1000 = 3$.
4. Raising a base to the logarithm recovers the number: $a^{\log_{a}N} = N$. This "undoing" property is simply the definition read the other way round.
5. The logarithm of zero and of negative numbers is undefined (in the real number system). No real power of a positive base $a$ can ever equal $0$ or a negative value, so $\log_{a}0$ and $\log_{a}(-5)$, for instance, do not exist. As $N$ shrinks towards $0$, $\log_{a}N$ falls without limit towards $-\infty$.
6. The base must be positive and not equal to 1. A base of 1 would give $1^{x}=1$ always, and a negative base would not give a consistent real power for fractional exponents — so both are excluded, as first noted in §1.
7. Logarithms of numbers between 0 and 1 are negative, since a fraction is a base raised to a negative power (e.g. $\log_{10}0.1 = \log_{10}(10^{-1}) = -1$), while logarithms of numbers greater than 1 (to a base greater than 1) are positive. …