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

Indicate the characteristics of a logarithmic function
Key Points

The inverse of the logarithmic operation is exponentiation.

The logarithm is commonly used in many fields: that with base 2 in computer science, that with base e in pure mathematics, and that with base 10 in natural science and engineering.

The logarithm of a product is the sum of the logarithms of the factors.
Terms

exponentiation
The process of calculating a power by multiplying together a number of equal factors, where the exponent specifies the number of factors to multiply.

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

exponent
The power to which a number, symbol, or expression is to be raised. For example, the 3 in
$x^3$ .
Full Text
In its simplest form, a logarithm is an exponent. Taking the logarithm of a number, one finds the exponent to which a certain value, known as a base, is raised to produce that number once more.
Logarithms have the following structure:
where b is known as the base, c is the exponent to which the base is raised to afford x.
Consider the following logarithm, for example:
The left side of the equation states that the right will be the exponent to which 3 is raised to yield 243 and indeed, 3^{5}=243.
The explanation of the previous example reveals the inverse of the logarithmic operation: exponentiation. 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 are most commonly used to simplify complex calculations that involve highlevel exponents. In science and engineering, the logarithm with base 10 is commonly used.
In pure mathematics, the logarithm with base e (≈ 2.718) is often applicable. In computer science, the binary logarithm, with base 2, is commonly used.
In natural science and engineering, the logarithm with base 10 is frequently used. In chemistry, for example, pH and pKa are used to simplify concentrations and dissociation constants, respectively, of high exponential value.
The purpose is to bring wideranging values into a more manageable scope. A dissociation constant may be smaller than 10^{10}, or higher than 10^{50}. Taking the logarithm of each brings the values into a more comprehensible scope (10 to 50) .
Key Term Reference
 base
 Appears in this related concepts: Logarithms of Powers, Balancing Redox Equations, and Strong Bases
 complex
 Appears in this related concepts: The ComplexNumber System, Electron Transport Chain, and Basic Techniques in Protein Analysis
 constant
 Appears in this related concepts: Inverse Variation, Quadratic Functions of the Form f(x) = a(xh)^2 + k, and Direct Variation
 e
 Appears in this related concepts: Derivatives of Exponential Functions, Derivatives of Logarithmic Functions, and Natural Logarithms
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 exponential
 Appears in this related concepts: Exponential Growth and Decay, Solving General Problems with Logarithms and Exponents, and Population Growth
 function
 Appears in this related concepts: Solving Differential Equations, Four Ways to Represent a Function, and Average Value of a Function
 logarithmic function
 Appears in this related concepts: Limited Growth and Graphs of Logarithmic Functions
 operation
 Appears in this related concepts: Designing the Operation, Integer Exponents, and Converting between Exponential and Logarithmic Equations
 pH
 Appears in this related concepts: AcidBase Indicators, Chemical Buffer Systems, and Calculating Changes in a Buffer Solution
Sources
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Cite This Source
Source: Boundless. “Logarithmic Functions.” Boundless Algebra. Boundless, 03 Jul. 2014. Retrieved 21 Apr. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/exponentsandlogarithms5/graphinglogarithmicfunctions37/logarithmicfunctions1745869/