logarithm

Examples of logarithm in the following topics:

• Complex Logarithms

• Logarithmic Functions

  • In its simplest form, a logarithm is an exponent.
  • A logarithm with a base of $10$ is called a common logarithm and is denoted simply as $logx$.
  • A logarithm with a base of $e$ is called a natural logarithm and is denoted $lnx$.
  • A logarithm with a base of $2$ is called a binary logarithm.
  • Starting with $243$, if we take its logarithm with base $3$, then raise $3$ to the logarithm, we will once again arrive at $243$.

• Bases Other than e and their Applications

  • These are $b = 10$ (common logarithm); $b = e$ (natural logarithm), and $b = 2$ (binary logarithm).
  • In this atom we will focus on common and binary logarithms.
  • The major advantage of common logarithms (logarithms in base ten) is that they are easy to use for manual calculations in the decimal number system:
  • Binary logarithm ($\log _2 n$) is the logarithm in base $2$.
  • For example, the binary logarithm of $1$ is $0$, the binary logarithm of $2$ is 1, the binary logarithm of $4$ is $2$, the binary logarithm of $8$ is $3$, the binary logarithm of $16$ is $4$, and the binary logarithm of $32$ is $5$.

• Logarithms of Products

  • A useful property of logarithms states that the logarithm of a product of two quantities is the sum of the logarithms of the two factors.
  • It is useful to think of logarithms as inverses of exponentials.
  • Logarithms were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily by using slide rules and logarithm tables.
  • Taking the logarithm base $b$ of both sides of this last equation yields:
  • Relate the product rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of products

• Natural Logarithms

  • The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
  • The natural logarithm is the logarithm with base equal to e.
  • The first step is to take the natural logarithm of both sides:
  • Using the power rule of logarithms it can then be written as:
  • The graph of the natural logarithm lies between the base 2 and the base 3 logarithms.

• Common Bases of Logarithms

  • While any positive number can be used as the base of a logarithm, not all logarithms are equally useful in practice.
  • A logarithm with a base of $10$ is called a common logarithm and is denoted simply as $logx$.
  • A logarithm with a base of $e$ is called a natural logarithm and is denoted $lnx$.
  • A logarithm with a base of $2$ is called a binary logarithm and is denoted $ldn$.
  • Logarithms are related to musical tones and intervals.

• Logarithms of Powers

  • The logarithm of the $p\text{th}$ power of a quantity is $p$ times the logarithm of the quantity.
  • We have already seen that the logarithm of a product is the sum of the logarithms of the factors:
  • Relate the power rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of powers

• Simplifying Expressions of the Form log_a a^x and a(log_a x)

  • The logarithm of the p-th power of a number is p times the logarithm of the number itself:
  • Similarly, the logarithm of a p-th root is the logarithm of the number divided by p:
  • Because $\log_a{a}=1$, the formula for the logarithm of a power says that for any number x:
  • This formula says that first taking the logarithm and then exponentiating gives back x.
  • Therefore, the logarithm to base-a is the inverse function of

• Solving General Problems with Logarithms and Exponents

  • Logarithms are useful for solving equations that require an exponential term, like population growth.
  • For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10 10 10 = 103.
  • Logarithms have several applications in general math problems.
  • We can rewrite logarithm equations in a similar way.
  • If you are asked to rewrite that logarithm equation as an exponent equation, think about it this way.

• Logarithms of Quotients

  • The logarithm of the ratio of two quantities is the difference of the logarithms of the quantities.
  • We have already seen that the logarithm of a product is the sum of the logarithms of the factors:
  • Similarly, the logarithm of the ratio of two quantities is the difference of the logarithms:
  • Relate the quotient rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of quotients