Laws of Logs

3 Key excerpts on "Laws of Logs"

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  AP Precalculus Premium, 2024: 3 Practice Tests + Comprehensive Review + Online Practice

  • Christina Pawlowski-Polanish(Author)
  • 2023(Publication Date)
  • Barrons Educational Services

    (Publisher)

  x = 0, with an end behavior that is unbounded. That is, for a logarithmic function in general form,

  7.4 Logarithmic Function Manipulation

  Since a logarithm is an exponent, the laws for evaluating logarithms follow the laws of exponents listed in Section 5.2. Logarithms of products, quotients, powers, and roots

  can be broken down into their component parts by using one or more of the logarithm laws outlined in Table 7.6 .

  Table 7.6 Logarithm Laws

  Example

  Express in terms of log5 x and log5 y.

  Solution

  First use the power law: Then use the quotient law: Therefore:

  Example

  If log 2 = x and log 3 = y, express each of the following in terms of x and y.

  1.

  2.log 24 3.log 1.5

  Solution

  For each of the examples, rewrite the numbers as factors of 2 and 3 and then use the logarithm laws.

  1.

  2.

  3.

  Each of these logarithm laws has a different graphical implication. These graphic properties are outlined in Table 7.7 .

  Table 7.7 Logarithm Graphical Transformations

  Example

  Describe the transformation of each of the following functions from its parent function.

  1.f(x) = log2 2x

  2.g(x) = log3 x2

  3.h(x) = log5 x

  4.

  Solution

  For each of the functions, use the logarithmic laws to rewrite each function as a form of the parent function y = a log

  b

  x. Then use the differences to describe the transformations.

  1.f(x) = log2 2x = log2 2 + log2 x = 1 + log2 x

  When written in this form, f(x) is a vertical translation 1 unit up of the graph of the function

  y = log2 x.

  2.g(x) = log3 x2 = 2 log3 x

  When written in this form, g(x) is a vertical dilation of the function y = log3 x by a factor of 2.

  3.

  When written in this form, h(x) is a vertical dilation of the function y = log5 x by a factor of

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  A Journey into the World of Exponential Functions

  • Gautam Bandyopadhyay(Author)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  Main impetus in this regard came from astronomy where it was frequently necessary to multiply and divide large numbers. However, logarithm can be perceived from many other angles. It can be viewed as the area under the rectangular hyperbola y = 1 x in geometry. It can be used as the inverse of exponential function e x or a x. As such we may treat it as the inverse of continuous compounding problem when we are interested to know in how many years Rs. 1/- will have a matured value e x or a x. In analysis we find that it is the limit of the product of two factors which are functions of n when n tends to infinity. It can also be expressed as an infinite series. It is one of the core functions in mathematics extended to negative and complex numbers. It plays vital roles in many branches of mathematics. Mathematical expressions for inductance and capacitance of a transmission line contain logarithmic terms. Logarithm forms the basis of Richter scale and measure of pH. It has wide applications in many other fields as well. 3.2 Logarithm as artificial numbers facilitating computation “Logarithms are a set of artificial numbers invented and formed into tables for the purpose of facilitating arithmetical computations. They are adapted to the natural numbers in such a manner that by their aid Addition supplies the place of Multiplication, Subtraction to that of Division, Multiplication that of Involution, and Division that of Evolution or the Extraction of Roots”. Excerpt from A Manual of Logarithms and Practical Mathematics for the use of students, Engineers, Navigators and Surveyors — by James Trotter of Edinburgh Published by Oliver & Boyd, Tweeddale Court and Simpkin, Marshall, & Co. London in 1841. In eleventh century Ibon Jonuis, an Arab mathematician proposed a method of multiplication which can save computational labour significantly. The method is known as Prosthaphaeresis. The Greek word prosthesis means addition and aphaeresis means subtraction

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  Calculus: The Logical Extension of Arithmetic

  • Seymour B. Elk(Author)
  • 2016(Publication Date)
  • Bentham Science Publishers

    (Publisher)

  In Chapter 2, the properties of logarithms were developed for any base, but primarily when the base of the numbering system was the number 10. Such logarithms were designated with the adjective “common” in conformity with the basis of our number system being the biological “accident” that the human species has evolved to have ten fingers. Meanwhile, a similar system, called “natural logarithms”, in which a different base number designated as e, will be shown to have mathematical importance. This new variety of logarithm and this special number e are encountered in diverse fields such as science and economics. The definition of e will be in terms of a definite integral with the argument of the function being one, or both, of its limits and the integrand being a “dummy variable”. The added perspective that will accompany this fundamental constant of both nature and of advanced mathematics, e, will be a third member of the set of indeterminate forms, cataloged by l'Hôpital in 1696, which was discussed, but not then assigned a name other than L 7 in Section 3.2; namely one raised to the infinite power (1 ∞). Next, employing algebraic properties associated with exponents, a pragmatic technique, which is applicable to functions that are combinations of multiplication, division, exponents and roots, while simultaneously being limited with respect to addition and subtraction will be described. This technique, called logarithmic differentiation, has been devised so as to decrease the tedium of selected traditional differentiations in many instances by employing properties of algebra. In a similar manner, this number e, as the base of the function e x, has the important functional identity property that its derivative is equal to the function itself. Moreover, every higher derivative (and integral, when the constants of integration are set equal to zero) is also equal to this function

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