logarithmic function

(noun)

Any function in which an independent variable appears in the form of a logarithm. The inverse of a logarithmic function is an exponential function and vice versa. 

Related Terms

Examples of logarithmic function in the following topics:

  • 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.
    • The natural logarithm is the inverse of the exponential function $f(x)=e^x$.
    • The graph of the logarithm function $log_b(x)$ (blue) is obtained by reflecting the graph of the function $b(x)$ (red) at the diagonal line ($x=y$).

  • Graphs of Logarithmic Functions

    • At first glance, the graph of the logarithmic function can easily be mistaken for that of the square root function.
    • When graphing without a calculator, we use the fact that the inverse of a logarithmic function is an exponential function.
    • 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.

  • 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 $243$.

  • Restricting Domains to Find Inverses

    • Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.
    • This is not true of the function $f(x)=x^2$.
    • Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.  
    • The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
    • No, the function has no defined value for $x=0$. 

  • Natural Logarithms

    • The natural logarithm is the logarithm with base equal to e.
    • Just as the exponential function with base $e$ arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base $e$, also arises in naturally in many contexts.
    • It is used much more frequently in physics, chemistry, and higher mathematics than other logarithmic functions.
    • The natural logarithm function can be used to solve equations in which the variable is in an exponent.
    • The graph of the natural logarithm lies between the base 2 and the base 3 logarithms.

  • Solving Problems with Logarithmic Graphs

    • Some functions with rapidly changing shape are best plotted on a scale that increases exponentially, such as a logarithmic graph.
    • Thus, it becomes difficult to graph such functions on the standard axis.
    • For very steep functions, it is possible to plot points more smoothly while retaining the integrity of the data: one can use a graph with a logarithmic scale .
    • Here are some examples of functions graphed on a linear scale, semi-log and logarithmic scales.
    • That means that if we want to graph a function that is unwieldy on a linear scale we can use a logarithmic scale on each axis and retain the properties of the graph while at the same time making it easier to graph.

  • Introduction to Inverse Functions

    • Below is a mapping of function $f(x)$  and its inverse function, $f^{-1}(x)$.  
    • In general, given a function, how do you find its inverse function?
    • As soon as the problem includes an exponential function, we know that the logarithm reverses exponentiation.  
    • The complex logarithm is the inverse function of the exponential function applied to complex numbers.  
    • A function's inverse may not always be a function.  

  • 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

  • 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