natural logarithm

(noun)

The logarithm in base e.

Related Terms

Examples of natural logarithm in the following topics:

  • 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 natural logarithm can be written as $\log_e x$ but is usually written as $\ln x$.
    • The first step is to take the natural logarithm of both sides:
    • The graph of the natural logarithm lies between the base 2 and the base 3 logarithms.

  • Common Bases of Logarithms

    • In the present day, logarithms have many uses in disciplines as different as economics, computer science, engineering and natural sciences.
    • Common logarithms are often used in physical and natural sciences and engineering.
    • A logarithm with a base of $e$ is called a natural logarithm and is denoted $lnx$.
    • Natural logarithms are also used in physical sciences and pure math.
    • Natural logarithms are closely linked to counting prime numbers ($2, 3, 5, 7$ ...), an important topic in number theory.

  • The Number e

    • When used as the base for a logarithm, we call that logarithm the natural logarithm and write it as $\ln x$.
    • The number $e$, sometimes called the natural number, or Euler's number, is an important mathematical constant approximately equal to 2.71828.
    • When used as the base for a logarithm, the corresponding logarithm is called the natural logarithm, and is written as $\ln (x)$.

  • 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 $f(x)=e^x$.

  • 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 $lnx$.
    • The irrational number  $e\approx 2.718 $ arises naturally in financial mathematics, in computations having to do with compound interest and annuities.
    • A logarithm with a base of $2$ is called a binary logarithm.

  • Converting between Exponential and Logarithmic Equations

    • Starting with $243$, if we take its logarithm with base $3$, then raise $3$ to the logarithm, we will once again arrive at $243$.
    • While you can take the log to any base, it is common to use the common log with a base of $10$ or the natural log with the base of $e$.
    • This is because scientific and graphing calculators are equipped with a button for the common log that reads $log$, and a button for the natural log that reads $ln$ which allows us to obtain a good approximation for the common or natural log of a number.
    • Solve for $x$ in the equation $2^x=17$ using the natural log
    • Here we will use the natural logarithm instead to illustrate the fact that any base will do.

  • Graphs of Logarithmic Functions

    • This means the point $(x,y)=(1,0)$ will always be on a logarithmic function of this type.
    • When graphing with a calculator, we use the fact that the calculator can compute only common logarithms (base is $10$), natural logarithms (base is $e$) or binary logarithms (base is $2$).
    • Thus far we have graphed logarithmic functions whose bases are greater than $1$.
    • The graph of the logarithmic function with base $3$ can be generated using the function's inverse.
    • Its shape is the same as other logarithmic functions, just with a different scale.

  • Complex Logarithms

  • 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

  • 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