logarithmic function

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.

• Derivatives of Logarithmic Functions

  • The general form of the derivative of a logarithmic function is $\frac{d}{dx}\log_{b}(x) = \frac{1}{xln(b)}$.
  • Here, we will cover derivatives of logarithmic functions.
  • Next, we will raise both sides to the power of $e$ in an attempt to remove the logarithm from the right hand side:
  • We can use the properties of the logarithm, particularly the natural log, to differentiate more difficult functions, such as products with many terms, quotients of composed functions, or functions with variable or function exponents.
  • We do this by taking the natural logarithm of both sides and re-arranging terms using the following logarithm laws:

• Exponential and Logarithmic Functions

  • Both exponential and logarithmic functions are widely used in scientific and engineering applications.
  • Exponential function is the function $e^x$ the number (approximately 2.718281828) such that the function $e^x$ is its own derivative .
  • The logarithm to base $b=10$ is called the common logarithm and has many applications in science and engineering.
  • The binary logarithm uses base $b=2$ and is prominent in computer science.
  • The derivative (or slope of a tangential line) of the exponential function is equal to the value of the function.

• The Natural Logarithmic Function: Differentiation and Integration

  • Differentiation and integration of natural logarithms is based on the property $\frac{d}{dx}\ln(x) = \frac{1}{x}$.
  • The natural logarithm, generally written as $\ln(x)$, is the logarithm with the base e, where e is an irrational and transcendental constant approximately equal to $2.718281828$.
  • The natural logarithm allows simple integration of functions of the form $g(x) = \frac{f '(x)}{f(x)}$: an antiderivative of $g(x)$ is given by $\ln\left(\left|f(x)\right|\right)$.

• 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 $243$.

• 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.

• 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.
  • Binary logarithm ($\log _2 n$) is the logarithm in base $2$.
  • It is the inverse function of $n \Rightarrow 2^n$.
  • 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 $5$.

• 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