Logarithm

Computing logarithms is the inverse of exponentiation, as subtraction is the inverse of addition and division is the inverse of multiplication. For example, since $343=7^{3}$, then the base-7 logarithm of 343 is 3. In general, if $a=b^{c}$, then $\log _{b}(a)=c$.

Because we use a base-10 number system, it is often convenient to use 10 as the base of the logarithm. Multiplying a number by 10 appends a zero to the numeral and adds 1 to the logarithm. For example, $\log _{10}(43)$ is approximately equal to 1.63347, and multiplying by 10, $\log _{10}(430)$ is approximately 2.63347.

Mathematicians and physicists often find, however, that the transcendental number $e$ is more convenient as a base for a logarithmic function. The value of $e$ is approximately equal to 2.718281828459045. The logarithmic function with $e$ as a base is called the "natural logarithm" and is often written $\ln(x)$; it is the function whose derivative is $1/x$.

Logarithms of intermediate and fractional values

Considering logarithms to the base 10, it's easy to find the logarithm of a number such as 1000, because $1000=10^{3},$ therefore $log_{10}(1000)=3.$ (The value of the logarithmic function is the number that appeared as an exponent in the other equation.) But what is the logarithm of a number such as 43, or 0.001?

We know from the study of exponents that, for example, $10^{\frac {1}{2}}$ is ${\sqrt {10}}$ and this then supplies a value for $\log _{10}({\sqrt {10}})=0.5.$ Values for many other numbers can be worked out similarly using cube roots and so on, and values for all real numbers can then be defined using limits.

$10^{-3}={\frac {10}{10^{4}}}=0.001$

and it then follows that

$\log _{10}(0.001)=-3,$

or in general,

$\log _{b}\left({\frac {1}{x}}\right)=-\log _{b}(x).$

By a similar argument it can be established that $b^{0}=1$ for any base $b>1$ and therefore that $\log _{b}(1)=0.$

Thus, logarithms of numbers between 0 and 1 are negative numbers, the logarithm of 1 is 0, and logarithms of numbers that fall between powers of the base are (usually non-integer) real numbers.

Shape of the logarithm function

Consider the function $f(x)=\log _{b}(x)$ where b is a real-number base greater than 1 of the logarithm. Since any positive real number raised to the exponent zero is 1, the logarithm of 1 is zero. To the right of 1 on the x-axis, the function $f$ continually increases, but increases more and more slowly as $x$ heads towards infinity. Between 1 and 0, the logarithmic function has negative values, and asymptotically approaches minus infinity as $x$ approaches zero. For negative values of $x$, there is no defined value of $f(x)$ within the real numbers — but using complex numbers a value can be found, as will be discussed further below.

Manipulating logarithms

Suppose we know the logarithm of a number using one base $b$, and we want to find the logarithm using a different base $c$ :

$\log _{b}(x)=y$

$\log _{c}(x)=$?

Suppose we know $y$, $b$ and $c$ and we want to find $\scriptstyle {\log _{c}(x)}$. We need to look up the value of $\log _{c}(b)$, and then by multiplying we can find the desired quantity:

$\log _{c}(x)=\log _{c}(b)\log _{b}(x)$

This formula can be established using the definition of logarithms and the rule for multiplying exponents:

$x=b^{\log _{b}x}$

$b=c^{\log _{c}b}$

Substituting for b,

$x=(c^{\log _{c}b})^{\log _{b}x}$

$x=c^{\log _{c}(b)\log _{b}(x)}$

Therefore $\log _{c}(x)=\log _{c}(b)\log _{b}(x)$, the formula we wished to prove.

A useful formula for $\log(xy)$ can be derived using the rule for adding exponents:

$b^{\log _{b}(xy)}=xy=b^{\log _{b}x}b^{\log _{b}y}=b^{\log _{b}x+\log _{b}y}$

Therefore :

$\log _{b}(xy)=\log _{b}(x)+\log _{b}(y)$

Notational variants

When logarithms are used to measure physical values (e.g., when noise is measured in deciBels) it is common to use 10 as a base, and in these contexts, the notation $\scriptstyle \log x$ means $\scriptstyle \log _{10}\,x$. In mathematics, it is more useful to use natural logarithms (i.e., logarithms to the base of $e$). The notation $\scriptstyle \ln x$ is sometimes used for natural logarithms, but in mathematics it is just as common to use the notation $\log \,x$ and write logarithms to base 10 using the full notation $\scriptstyle \log _{10}\,x$. Computer science adds a whole new wrinkle because, in some cases, it is most useful to consider logarithms to base 2, and this has led to a new notation $\mathrm {lg} \,x$ which become common for logarithms to a base of 2. Finally, the notation $\scriptstyle \log \,x$ is sometimes used for logarithms to base 2, but this is usage is less common.

Complex numbers and logarithms

The exponential function of an imaginary number is given as

$e^{ix}=\cos(x)+i\sin(x).\,$

To find the logarithm of a complex number $a+bi$, it's convenient to express the number in polar coordinates $(r,\theta )$ where $\scriptstyle r={\sqrt {a^{2}+b^{2}}}$ and $\theta =\arctan(b/a)$. (Intuitively, $r$ is the length of the line segment joining the number to the origin in the complex plane, and $\theta$ is the angle that line segment makes with the $x$-axis.) The equivalence of the two notations is given by

$a+bi=r(\cos(\theta )+i\sin(\theta )).\,$

Suppose we define $c+di$ such that $\ln(a+bi)=c+di$. Using the formula for the exponential function above,

$a+bi=e^{c+di}=e^{c}e^{di}=e^{c}(\cos(d)+i\sin(d))\,$

It can be seen from similarity with the above formula for polar coordinates that $\scriptstyle {r=e^{c}}$ and $\scriptstyle {d=\theta }$. Therefore,

$\ln(a+bi)=c+di=\ln(r)+i\theta =\ln({\sqrt {a^{2}+b^{2}}})+i\arctan \left({\frac {b}{a}}\right).\,$

In this way, the logarithmic function can be extended to cover the entire complex plane except for the number zero, which has an undefined value — a singularity with the real part plunging towards minus infinity and the imaginary part spinning wildly.