logarithm

(noun)

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.

Related Terms

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 $243$.

  • 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

  • Introduction to Exponential and Logarithmic Functions

    • Logarithmic functions and exponential functions are inverses of each other.
    • The inverse of an exponential function is a logarithmic function and vice versa.
    • In the following graph you can see an exponential function in red and its inverse, a logarithmic function, in blue.
    • Let us consider instead the natural log (a logarithm of the base $e$). 
    • The natural logarithm is the inverse of the exponential function $f(x)=e^x$.