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Topic: Logarithmic identities

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Logarithm

E (mathematical constant)

Geometric mean

St Albans Cathedral

Khalil Gibran

	Logarithm Summary
		Logarithms with 10 as a base are called " common" or " Briggsian." The other exception is when the base is the number e (which equals 2.718282...).
		Since tables of logarithms show positive mantissas only, a logarithm such as -5.8111 must be converted to.1889 - 6 before a table can be used to find the "antilogarithm," which is the name given to the number whose logarithm it is. A calculator will show the antilogarithm without such a conversion.
		The logarithms he invented, however, were not the simple logarithms we use today (his logarithms were not what are now called "Napierian"), Shortly after Napier published his work, Briggs, an English mathematician met with him and together they worked out logarithms that much more closely resemble the common logarithms that we use today.
	www.bookrags.com /Logarithm (4345 words)

	Reference.com/Encyclopedia/Logarithmic identities
		Logarithms can be used to make calculations easier.
		This identity is needed to evaluate logarithms on calculators.
		The identities of logarithms can be used to approximate large numbers.
	www.reference.com /browse/wiki/Logarithmic_identities (480 words)

	Logarithm (Site not responding. Last check: 2007-11-03)
		Logarithms tell how many times a number x must be divided by the base b to get 1, and hence can be considered an inverse of exponentiation.
		Logarithms convert multiplication to addition, division to subtraction (making them isomorphisms between the field operations), exponentiation to multiplication, and roots to division (making Logarithms crucial to slide rule construction).
		Logarithms are useful in order to solve equations in which the unknown appears in the exponent, and they often occur as the solution of differential equations because of their simple derivatives.
	logarithm.iqnaut.net (1626 words)

	Solving Logarithmic Equations (Site not responding. Last check: 2007-11-03)
		Logarithms convert multiplication to addition, division to subtraction (makingthem isomorphisms between the field operations), exponentiation to multiplication, and roots to division (making them crucial to slide rule construction).
		Usage Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1.7.
		Logarithms are useful in order to solve equations in which the unknown appears in the exponent, and they often occur as thesolution of differential equations because of theirsimple derivatives.
	www.beyondtheorange.com /Help/2763-Solving-Logarithmic-Equations.Html (603 words)

	Logarithmic Functions
		There are several properties of logarithmic functions that follow easily from the definition and are evident from the graphs in the applet above.
		It follows that its inverse, the logarithm with base e, is the most important of the logarithmic functions.
		The logarithm with base e is called the natural logarithm, and it is denoted ln.
	www.uncwil.edu /courses/mat111hb/EandL/log/log.html (673 words)

	Review Sheet for Midterm 2
		Sketch a graph of a logarithmic function.Â Specify the domain and range of a given logarithmic function.Â Convert an expression involving a logarithmic function into an equivalent expression involving an exponential function (i.e.
		Solve equations involving logarithms, by converting logarithmic equations to exponential equations.Â Solve equations with variables in exponents by simplifying and taking the log of both sides of the equation.Â Solve inequalities using logarithms and exponents.Â (Remember to intersect the solution set you obtain with the domain of the original equation to obtain your final answer).
		Specify the location of angles measured in radians on the unit circle.Â Convert degrees to radians and vice versa.Â State and apply the relationship between radian measure of an angle and arc length on a circle with a given radius.
	math.berkeley.edu /~benjamin/review_2.htm (871 words)

	Algebra II: Exponential and Logarithmic Functions - Math for Morons Like Us
		Logarithms are used all the time in real life, for example, the Richter Scale, and are very useful for measuring things that grow or diminish exponentially.
		Logarithmic functions are the inverse of exponential functions.
		This definition is explained by knowing how to convert exponential equations to logarithmic form, and logarithmic equations to exponential form.
	library.thinkquest.org /20991/alg2/log.html (572 words)

	aw_rockswold_precalculu_2\|College Algebra & Trigonometry through M\|Study Tips
		Notice that the output from the common logarithmic function is an exponent of a power of 10.
		Identities are often used in mathematics to simplify expressions.
		Identities are typically a new topic for most students and require extra study time to master.
	wps.aw.com /aw_rockswold_precalculu_2/0,1853,52365-,00.html (5906 words)

	Logarithmic identities
		What follows is a list of identities that are useful when dealing with logarithms.
		All of these are valid for all positive real numbers a, b and c except that the base of a logarithm may never be 1.
		The last limit is often summarized as "logarithms grow more slowly than any power or root of x".
	publicliterature.org /en/wikipedia/l/lo/logarithmic_identities.html (146 words)

	Review of Trigonometric, Logarithmic, and Exponential Functions - HMC Calculus Tutorial
		In this tutorial, we review trigonometric, logarithmic, and exponential functions with a focus on those properties which will be useful in future math and science applications.
		Logarithmic and exponential functions are inverses of each other:
		Notice that each curve is the reflection of the other about the line y = x.
	www.math.hmc.edu /calculus/tutorials/reviewtriglogexp (100 words)

	CPMP Course 4 Units
		Inverses of functions; logarithmic functions and their relation to exponential functions, properties of logarithms, equation solving with logarithms; logarithmic scales and re-expression, linearizing data, and fitting models using log and log-log transformations.
		Extends student ability to manipulate symbolic representations of exponential, logarithmic, and trigonometric functions; to solve exponential and logarithmic equations; to prove or disprove that two trigonometric expressions are identical and to solve trigonometric equations; to reason with complex numbers and complex number operations using geometric representations and to find roots of complex numbers.
		Definition of e and natural logarithms; the tangent, cotangent, secant, and cosecant functions; use of substitution and extraneous solutions in equation solving; methods of solving radical, rational, exponential, logarithmic, and logistic equations; solving trigonometric equations in exact form including expression of periodic solutions; trigonometric identities including sum and difference identities, double- and half-angle identities.
	www.wmich.edu /cpmp/course4.html (921 words)

	Logarithm - Wikipedia, the free encyclopedia
		Logarithms of all bases pass through the point (1, 0), because any number raised to the power 0 is 1, and through the points (b, 1) for base b, because any number raised to the power 1 is itself.
		The derivative of the natural logarithm function is easily found via the inverse function rule.
		François Callet's seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of interpolation, which were greatest in the early part of the table; and this addition was generally included in seven-place tables.
	en.wikipedia.org /wiki/Logarithm (3078 words)

	FCPS Instructional Services
		The course also includes the study of limits, continuity, maximum and minimum points and values, definition and properties of the derivative, rules of differentiation, equations of tangent lines to polynomial functions, infinite limits, and partial fractions.
		This will include the role of e, natural logarithms, common logarithms, laws of exponents, and solutions of logarithmic and exponential equations.
		The student will use the sum and difference identities and the half-angle and double angle identities to verify other trigonometric identities and solve equations.
	www.fcps.k12.va.us /DIS/pos/math/precalculush316036.htm (2749 words)

	Geometric mean (Site not responding. Last check: 2007-11-03)
		By using logarithmic identities to transform the formula, we can express the multiplications as a sum and the power as a multiplication.
		I.e., it is the generalised f-mean with f(x) = ln x Therefore the geometric mean is related to the log-normal distribution.
		The log-normal distribution is a distribution which is normal for the logarithm transformed values.
	geometric-mean.iqnaut.net (355 words)

	MATH 147 006 - Review Notes for Test #3 -- 4/1/05
		Be prepared to use your calculator in connection with logarithms and exponential models of growth and decay.
		Decoding the APR depending on how many time in one year the interest is compounded.
		Know what a logarithm is and how to solve exponential equations, and you won't need to burden your memory with BLUE362.
	math.boisestate.edu /~kerr/147sp05/review_suggestions_3.shtml (650 words)

	MAT114 Elementary Functions
		They should be able to solve exponential and logarithmic equations and should be able to solve applied problems involving growth and decay.
		Students should know the definitions and basic identities of trigonometric functions and should be able to solve applied trigonometry problems.
		Topics to be covered include exponential and logarithmic functions, complex numbers and polynomial functions, trigonometry, plane analytic geometry, and systems of linear equations and inequalities.
	www.assumption.edu /users/kcarlin/syllabi/MAT114.html (457 words)

	Honors Pre-Calc 220 Chapter Objectives
		Apply the Change-of-Base formula to logarithms whose base is neither 10 or e.
		Use the fundamental trigonometric identities to determine the exact trigonometric value of an angle.
		Use trigonometric identities such as the sum/difference, double-angle, half-angle, product-to-sum, and sum-to-product identities to determine the exact trigonometric value of an angle.
	marian.creighton.edu /~dkath/courseobjectprecalc.htm (1175 words)

	Course 4 Unit 7 - Functions and Symbolic Reasoning
		The content of this unit is from both the algebra and functions strand and the geometry and trigonometry strand.
		Functions and Symbolic Reasoning extends student ability to manipulate symbolic representations of exponential, common and natural logarithmic, and trigonometric functions and to solve exponential, logarithmic, and trigonometric equations.
		Trigonometric identities are developed and proved or disproved.
	www.wmich.edu /cpmp/unitsamples/c4u7/c4u7intro.html (459 words)

	FVCC-MIS Course Descriptions - MATHEMATICS
		The course covers the topics of graphs of equations and inequalities, system of equations and inequalities, rational expressions and equations, radical expressions and equations, quadratic equations, exponential and logarithmic equations.
		This course consists of equations, systems of equations and methods of solution, exponents and radicals, linear and quadratic functions and their graphs, linear programming, exponential and logarithmic functions, sequences and series, induction, and the binomial expansion.
		Trigonometric and logarithmic functions, graphs, both circular and angular functions, identities, equations, applications.
	www.fvcc.edu /course.schedule/fvcc/TERM_03SP/MATH.shtml (1020 words)

	pH K-12 Experiments for Lesson Plans & Science Fair Projects
		In dilute solutions (like river or tap water) the activity is approximately equal to the concentration of the H
		denotes the base-10 logarithm, and pH therefore defines a logarithmic scale of acidity.
		by logarithmic identities, we then have the relationship:
	www.juliantrubin.com /encyclopedia/environment/ph.html (1378 words)

	Logarithmic identities - Wikipedia, the free encyclopedia
		It has been suggested that Change of base formula for logs be merged into this article or section.
		For instance, most calculators have buttons for ln and for log
		The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities:
	en.wikipedia.org /wiki/Logarithmic_identities (260 words)

	BT Research - PH (Site not responding. Last check: 2007-11-03)
		Though pH is generally expressed without units, it is not an arbitrary scale; the number arises from a definition based on the activity of hydrogen ions in the solution.
		The pH scale is a reverse logarithmic representation of relative hydrogen proton (H
		As the pH scale is logarithmic; it doesn't start at zero.
	breathittteens.com /research.php?title=PH (1327 words)

	Math 153 Description and Policies
		Exponential and logarithmic functions and equations; exponential growth and decay.
		obtain the domain and graph of linear, quadratic, exponential, logarithmic, trigonometric, and inverse trigonometric functions
		use a graphic calculator to graph and evaluate exponential, logarithmic, and trigonometric functions
	www.csudh.edu /math/smoite/Math153Spring2006Description.htm (539 words)

	Calculus@Internet
		SOLVING LOGARITHMIC EQUATIONS - To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable.
		Exponential and Logarithmic Functions - If f is a nonconstant function that is continuous and satisfies the functional equation f(x+y) = f(x) * f(y) then f(x) = ax for some constant a.
		Properties of Logarithms - One dilemma is that your calculator only has logarithms for two bases on it.
	www.calculus.net /ci2/search/?request=category&code=11141&off=0&tag=9200438920658 (351 words)

	SMC's Mathematics Department
		Topics include algebraic, exponential, logarithmic and trigonometric functions and their inverses and identities, conic sections, sequences, series, the binomial theorem, and mathematical induction.
		Analyze and graph a given function, including but not limited to piecewise defined, polynomial, rational, exponential, logarithmic, trigonometric, and inverse trigonometric functions.
		State and apply fundamental trigonometric identities and the sum, difference, double-angle and half-angle identities.
	www.smc.edu /math/course_info_math_02.htm (384 words)

	Course Outline
		It is expected that students will solve exponential, logarithmic and trigonometric equations and identities.
		use sum, difference,and double angle identities for sine and cosine to verify and simplify trigonometric expressions
		It is expected that students will represent and analyse exponential and logarithmic functions, using technology as appropriate.
	homepage.mac.com /pathill/school/Pages/outline.html (741 words)

	Math.Log Method (Double) (System)
		Returns the natural (base e) logarithm of a specified number.
		A number whose logarithm is to be found.
		The natural logarithm of d; that is, ln d, or log
	msdn2.microsoft.com /en-us/library/x80ywz41.aspx (610 words)

	PSU Undergraduate Mathematics Summer Review
		Manipulate logarithmic expressions, graph logarithmic functions and solve logarithmic equations
		Application of basic identities to the solution of trigonometric equations and proving identities
		The most appropriate texts for review of these fundamentals are those high school texts with which the student is already familiar.
	www.math.psu.edu /UG/summerreview.htm (371 words)

	Precalculus with Trigonometry
		Prerequisite: Advanced Algebra GT Precalculus With Trigonometry includes all the topics of Trigonometry (3150) and an in-depth treatment of functions through the study of polynomials, transformations, rational, exponential, and logarithmic functions, inverses, polar equations, parametric equations, two-dimensional vectors, and selected topics in discrete mathematics.
		Investigate and identify the characteristics of exponential and logarithmic functions in order to graph these functions, solve equations, and solve real-world problems.
		Use the sum and difference identities and the half-angle and double angle identities to verify other trigonometric identities and solve equations.*
	www.fcps.k12.va.us /sol/precalc.htm (963 words)

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