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

Practice working with logarithmic functions and identify their parts
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 financial mathematics, and that with base$10$ in natural science and engineering.
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.

exponent
The power to which a number, symbol, or expression is to be raised. For example, the
$3$ in$x^3$ . 
logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
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:
Note that
Commonly Used Bases
A logarithm with a base of
A logarithm with a base of
A logarithm with a base of
The Exponential and Logarithmic Forms of an Equation
Logarithmic equations can be written as exponential equations and vice versa. The logarithmic equation
As an example, the logarithmic equation
Example 1: Solve for
Here we are looking for the exponent to which
It might be more familiar if we convert the equation to exponential form giving us:
Thus,
The explanation of the previous example reveals the inverse of the logarithmic operation: exponentiation. Starting with
Trivial Logarithmic Identities
The following two logarithmic identities can be verified by converting the logarithmic equation into an exponential equation as follows:
Converting this to a logarithmic equation yields:
Converting
Applications of Logarithms
Historically, logarithms were invented by John Napier as a way of doing lengthy arithmetic calculations prior to the invention of the modern day calculator.
More recently, logarithms are most commonly used to simplify complex calculations that involve highlevel exponents. 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
Key Term Reference
 Interest
 Appears in these related concepts: Accounting for Interest Earned and Principal at Maturity, Interest Compounded Continuously, and Tax Considerations
 arithmetic
 Appears in these related concepts: Arithmetic Sequences, Sums and Series, and Sums, Differences, Products, and Quotients
 base
 Appears in these related concepts: Balancing Redox Equations, The Role of the Kidneys in AcidBase Balance, and Temple Architecture in the Greek Orientalizing Period
 complex
 Appears in these related concepts: Introduction to Complex Numbers, Electron Transport Chain, and Transduction of Sound
 compound interest
 Appears in these related concepts: Calculating Future Value, Calculating Present Value, and MultiPeriod Investment
 constant
 Appears in these related concepts: Inverse Variation, Graphing Quadratic Equations in Vertex Form, and Direct Variation
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 exponential
 Appears in these related concepts: Exponential Growth and Decay, Solving General Problems with Logarithms and Exponents, and Population Growth
 function
 Appears in these related concepts: Solving Differential Equations, Functions and Their Notation, and The Vertical Line Test
 indeterminate
 Appears in these related concepts: Indeterminate Forms and L'Hôpital's Rule, Calculating Limits Using the Limit Laws, and Basics of Graphing Polynomial Functions
 natural logarithm
 Appears in these related concepts: The Integral Test and Estimates of Sums, Natural Logarithms, and Common Bases of Logarithms
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