logarithm

Examples of logarithm in the following topics:

• 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 $5$.

• 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.
  • First, we will derive the equation for a specific case (the natural log, where the base is $e$), and then we will work to generalize it for any logarithm.
  • Next, we will raise both sides to the power of $e$ in an attempt to remove the logarithm from the right hand side:
  • 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.
  • The logarithm of a number $x$ with respect to base $b$ is the exponent by which $b$ must be raised to yield $x$.
  • In other words, the logarithm of $x$ to base $b$ is the solution $y$ to the equation $b^y = x$ .
  • 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.

• Logarithmic Functions

  • 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.
  • It follows that the logarithm of 8 with respect to base 2 is 3, so log2 8 = 3.
  • What would be the logarithm of ten?
  • The logarithm is denoted "logb(x)".

• 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 $\ln\left(\left|f(x)\right|\right)$.

• Further Transcendental Functions

  • Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.
  • Note that, for $ƒ_2$ in particular, if we set $c$ equal to $e$, the base of the natural logarithm, then we find that $e^x$ is a transcendental function.
  • Similarly, if we set $c$ equal to $e$ in ƒ5, then we find that $\ln(x)$, the natural logarithm, is a transcendental function.
  • One could attempt to apply a logarithmic identity to get $\log(10) + \log(m)$, which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.

• Integration Using Tables and Computers

  • Here are a few examples of integrals in these tables for logarithmic functions:

• Alternating Series

  • The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).

• Separable Equations

  • ., combine all possible terms, rewrite any logarithmic terms in exponent form, and express any arbitrary constants in the most simple terms possible).

• The Integral Test and Estimates of Sums

  • The harmonic series $\sum_{n=1}^\infty \frac1n$ diverges because, using the natural logarithm (its derivative) and the fundamental theorem of calculus, we get: