Logarithm

Logarithms and exponents are very closely related. In fact, they are inverse functions. This means that logarithms can be used to reverse the result of exponentiation and vice versa, just as addition can be used to reverse the result of subtraction. Thus, if we have $a^x = b$ , then taking the logarithm with base $a$ on both sides will give us $x=\log_a{b}$ .

We would read this as "the logarithm of b, base a, is x". For example, we know that $3^4=81$ . To express the same fact in logarithmic notation we would write $\log_3 81=4$ .

Conventions

Depending on the field, the symbol $\log$ without a base can have different meanings. Sometimes in high schools, the symbol is used to refer to a base 10 logarithm. Thus, $\log(100)$ can mean $\log_{10}(100)=2$ . In these contexts, the symbol $\ln$ (an abbreviation of the French "logarithme normal," meaning "natural logarithm") is introduced to refer to the logarithm base e, or natural logarithm. However, the choice of base 10 is arbitrary, and convenient only for computations in a base-10 number system. The natural logarithm, however, has many convenient mathematical properties, so practicing mathematicians often take the symbol $\log$ to mean the natural logarithm and do not use the symbol $\ln$ . (This is an example of conflicting mathematical conventions.) In addition, the notation $\lg$ is often used by combinatorists and computer scientists to refer to the logarithm base $2$ . Occasionally, the base of the logarithms is irrelevant.

Logarithmic Properties

We can use the properties of exponents to build a set of properties for logarithms.

We know that $a^x\cdot a^y=a^{x+y}$ . We let $a^x=b$ and $a^y=c$ . This also makes $a^{x+y}=bc$ . From $a^x = b$ , we have $x = \log_a{b}$ , and from $a^y=c$ , we have $y=\log_a{c}$ . So, $x+y = \log_a{b}+\log_a{c}$ . But we also have from $a^{x+y} = bc$ that $x+y = \log_a{bc}$ . Thus, we have found two expressions for $x+y$ establishing the identity:

$\log_a{b} + \log_a{c} = \log_a{bc}.$

Using the laws of exponents, one can derive and prove the following identities:

$\log_a b^n=n\log_a b$
$\log_a b+ \log_a c=\log_a bc$
$\log_a b-\log_a c=\log_a \frac{b}{c}$
$(\log_a b)(\log_c d)= (\log_a d)(\log_c b)$
$\frac{\log_a b}{\log_a c}=\log_c b$ (the change of base formula)

These formulas also have a number of common special cases:

$\log_{a}b=\frac{1}{\log_{b}a}$ (sometimes known as the inverse property of logarithms)
$\log_{a^n} b^n=\log_a b$
$\log_{1/a} b=-\log_a b$

Problems

Evaluate $(\log_{50}{2.5})(\log_{2.5}e)(\ln{2500})$ .

Evaluate $(\log_2 3)(\log_3 4)(\log_4 5)\cdots(\log_{2005} 2006)$ .

Simplify $\frac 1{\log_2 N}+\frac 1{\log_3 N}+\frac 1{\log_4 N}+\cdots+ \frac 1{\log_{100}N}$ where $N=(100!)^3$ .

Natural Logarithm

The natural logarithm is the logarithm with base e. It is usually denoted $\ln$ , an abbreviation of the French logarithme normal, so that $\ln(x) = \log_e(x).$ However, in higher mathematics such as complex analysis, the base 10 logarithm is typically disposed with entirely, the symbol $\log$ is taken to mean the logarithm base e and the symbol $\ln$ is not used at all. (This is an example of conflicting mathematical conventions.)

$\ln a$ can also be defined as the area under the curve $y=\frac{1}{x}$ between 1 and a, or $\int^a_1 \frac{1}{x}\, dx$ .

All logarithms are undefined in nonpositive reals, as they are complex. From the identity $e^{i\pi}=-1$ , we have $\ln (-1)=i\pi$ . Additionally, $\ln (-n)=\ln n+i\pi$ for positive real $n$ .

Problems

Introductory

Intermediate

The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r).$