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Difference between revisions of "Logarithm"

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Revision as of 19:53, 30 November 2007

This is an AoPSWiki Word of the Week for Nov 29-Dec 5

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. Typically, in mathematics through the level of calculus, the symbol is used to refer to a base 10 logarithm. Thus, $\log(100)$ means $\log_{10}(100)=2$. Usually, the symbol $\ln$ (an abbreviation of the French "logarithme normal," meaning "natural logarithm") is introduced to refer to the logarithm base e. 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.)


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, we can derive and prove the following identities:

  • $\log_a b^n=n\log_a b$
  • $\log_a b+ \log_a c=\log_a bc$ (Known as the product property.)
  • $\log_a b-\log_a c=\log_a \frac{b}{c}$
  • $(\log_a b)(\log_c d)= (\log_a d)(\log_c b)$
  • $\frac{\log_a b}{\log_a c}=\log_c b$
  • $\log_{a^n} b^n=\log_a b$

Try proving all of these as exercises.

Here are some other less useful log properties that follow from these previous ones.

  • $\log_{a}b=\frac{1}{\log_{b}a}$ (Known as the inverse property)
  • $(\log_{a}b)(\log_{b}c)=\log_{a}c$ (Known as the chain rule.)
  • $\log_{1/a} b=-\log_a b$

Problems

  1. Evaluate $(\log_{50}{2.5})(\log_{2.5}e)(\ln{2500})$.
  1. Evaluate $(\log_2 3)(\log_3 4)(\log_4 5)\cdots(\log_{2005} 2006)$.
  1. 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 of $a$ is $\log_e a=\ln a$. The function $f(x)=\ln x$ is the inverse of $f(x)=e^x$.

$\ln a$ 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 $(a,r).$

Source

Olympiad

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