Mathematics
Laws of Logs
The laws of logarithms are a set of rules that help simplify and manipulate logarithmic expressions. These laws include the product rule, quotient rule, and power rule, which allow for the expansion and condensation of logarithmic expressions. By applying these laws, logarithmic calculations can be streamlined and complex expressions can be simplified.
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eBook - PDF- James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
These properties give logarithmic functions a wide range of applications, as we will see in Sections 4.6 and 4.7. ■ Laws of Logarithms Since logarithms are exponents, the Laws of Exponents give rise to the Laws of Logarithms. LAWS OF LOGARITHMS Let a be a positive number, with a 2 1. Let A , B , and C be any real numbers with A 0 and B 0. Law Description 1. log a 1 AB 2 log a A log a B The logarithm of a product of numbers is the sum of the logarithms of the numbers. 2. log a a A B b log a A log a B The logarithm of a quotient of numbers is the difference of the logarithms of the numbers. 3. log a 1 A C 2 C log a A The logarithm of a power of a number is the exponent times the logarithm of the number. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 4.4 ■ Laws of Logarithms 391 Proof We make use of the property log a a x x from Section 4.3. Law 1 Let log a A u and log a B √ . When written in exponential form, these equations become a u A and a √ B Thus log a 1 AB 2 log a 1 a u a √ 2 log a 1 a u √ 2 u √ log a A log a B Law 2 Using Law 1, we have log a A log a ca A B b B d log a a A B b log a B so log a a A B b log a A log a B Law 3 Let log a A u . Then a u A , so log a 1 A C 2 log a 1 a u 2 C log a 1 a uC 2 uC C log a A ■ EXAMPLE 1 ■ Using the Laws of Logarithms to Evaluate Expressions Evaluate each expression.
No longer available |Learn more- Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
THE POWER RULE OF LOGARITHMS: For any base a > 0, a ≠ 1, and any exponent value n , we have that log a ( u n ) 5 n log a u . Naturally, we can envision situations in which the properties of logarithms might be used in conjunction within a single problem. As long as we keep in mind the precise requirements of the individual properties, we will find that this is not beyond our ability to master. EXAMPLE 11.20 Apply the properties of logarithms to fully expand the logarithmic expressions: log 5 4 5 x y ln 7 3 2 4 x w SOLUTION The key to successfully simplifying the first expression is to think “big picture” when looking at the argument of the logarithm. Observe that the expression is enclosed in parentheses and that it is a fraction of the form u v , where u 5 x 4 and v 5 5 y . Thus, the quotient rule of logarithms is applicable to this expression. Use of the quotient rule yields log 5 x 4 2 log 5 5 y , which we then attempt to simplify further. The first logarithm involves the variable x raised to a power, and thus the power rule of logarithms applies to this expression. The second logarithm has argument 5 y , which is a product, and thus the product rule of logarithms applies to that expression: log 5 4 5 x y 5 log 5 x 4 2 log 5 5 y by the quotient rule 5 4 log 5 x 2 (log 5 5 1 log 5 y ) by the power rule and the product rule 5 4 log 5 x 2 log 5 5 2 log 5 y by the distributive property Recall that use of the properties of logarithms is dictated by the maxim that if a rule can be used, then it must be used. When we consider the individual logarithms in this expression, we notice that the second logarithm, log 5 5, can be simplified using one of the fundamental properties of logarithms, log a a 5 1, and thus the expression simplifies once more to 5 4 log 5 x 2 1 2 log 5 y In the second logarithmic expression, the argument is, initially, an exponential expression, and thus the power rule of logarithms applies in this case.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
Even when multiplying and dividing most numbers it was easier with logs. Mathematics text books had pages of log tables at the back for this purpose. Sliding rules were also used for logarithmic calculations. One example where logs are still used in real life is to find the number of payments that you need to make on a loan or the time it will take to reach an investment goal. Consider the following: The logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. Therefore a logarithm is an unknown exponent that will be determined. For example: The logarithm of 100 to base 10 is 2, written as log 10 100 = 2. We know that 100 is 10 to the power of 2, therefore 100 = 10 × 10 = 10 2 . ∴ If 100 = 10 2 Then log 10 100 = 2 Definition of logarithms If x = a y then y is the logarithm of x to base a . ∴ y = log a x • a must be a positive real number and a ≠ 1; • y must be a positive number y = log a x is pronounced as ‘ y = the logarithm of x to base a ’. The two statements above express the inverse relationship, showing how an exponential equation is equivalent to a logarithmic equation: A logarithmic function can be ‘undone’ by an exponential function and vice versa. 18 Module 1 • Exponents and logarithms Consider the following examples. Write the following exponential equations in logarithmic form. Example 1 10 3 = 1 000 • called the exponential form ∴ log 10 1 000 = 3 • called the logarithmic form log 10 1 000 can be written as log 1 000 = 3. 1 000 is the number; 10 3 is the exponential expression and 3 is the logarithm (exponent). log 10 x is usually written as log x . That means if there is no base indicated you should assume that the base is 10. Example 2 3 2 = 9 ∴ log 3 9 = 2 • 3 2 = 9 ∴ log 3 9 = 2 exponent 3 2 = 9 ↔ log 3 9 = 2 base Example 3 4 2 = 16 ∴ log 4 16 = 2 • 4 2 = 16 ∴ log 4 16 = 2 Example 4 8 = 2 3 ∴ log 2 8 = 3 • 2 3 = 8 ∴ log 2 8 = 3
- 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
eBook - PDFAlgebra
Form and Function
- William G. McCallum, Eric Connally, Deborah Hughes-Hallett(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
A Logarithm Is an Exponent This is the most important thing to keep in mind about logarithms. When you are looking for the logarithm of a number, you are looking for the exponent to which you need to raise 10 to get that number. 7.1 INTRODUCTION TO LOGARITHMS 255 Example 2 Use the definition to find log 1000, log 100, log 10. Solution Since log is the power to which you raise 10 to get , we have log 1000 = 3 because 10 3 = 1000 log 100 = 2 because 10 2 = 100 log 10 = 1 because 10 1 = 10. In the previous example we recognized the number inside the logarithm as a power of 10. Some- times we need to use exponent laws to put a number in the right form so that we can see what power of 10 it is. Example 3 Without a calculator, evaluate the following, if possible: (a) log 1 (b) log √ 10 (c) log 1 100,000 (d) log 0.01 (e) log 1 √ 1000 (f) log(−10) Solution (a) We have log 1 = 0 because 10 0 = 1. (b) We have log √ 10 = 1∕2 because 10 1∕2 = √ 10. (c) We have log 1 100,000 = −5 because 10 −5 = 1 100,000 . (d) We have log 0.01 = −2 because 10 −2 = 0.01. (e) We have log 1 √ 1000 = − 3 2 because 10 −3∕2 = 1 ( 10 3 ) 1∕2 = 1 √ 1000 . (f) Since any power of 10 is positive, −10 cannot be written as a power of 10. Thus, log(−10) is undefined. In Example 3(f) we see that we cannot take the logarithm of a negative number. Also, log 0 is not defined since there is no power of 10 that equals zero. However, the value of a logarithm itself can be negative, as in Example 3(c), (d), and (e), and it can be zero, as in Example 3(a). For a number that is not an easy-to-see power of 10, you can estimate the logarithm by finding two powers of 10 on either side of it. Example 4 Estimate log 63. Solution We use the fact that 10 < 63 < 100. Since 10 1 = 10 and 10 2 = 100, we can say that 1 < log 63 < 2. In fact, using a calculator, we have log 63 = 1.799. The definition of a logarithm as an exponent means that we can rewrite any statement about logarithms as a statement about powers of 10.
eBook - PDFFunctions Modeling Change
A Preparation for Calculus
- Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
For example, log(10 6 ) = 6 and 10 log 6 = 6. In general, For any , log(10 ) = and for > 0, 10 log = . Example 4 Evaluate without a calculator: (a) log ( 10 8.5 ) (b) 10 log 2.7 (c) 10 log(+3) Solution Using log(10 ) = and 10 log = , we have: (a) log ( 10 8.5 ) = 8.5 (b) 10 log 2.7 = 2.7 (c) 10 log(+3) = + 3 You can check the first two results on a calculator. Properties of Logarithms In Section 4.3, we solved the exponential equation 100 ⋅ 2 = 337,000,000 graphically. We now use logarithms and their properties to do so. These properties are justified on page 168. 166 Chapter 5 LOGARITHMIC FUNCTIONS Properties of the Common Logarithm • By definition, = log means 10 = . • In particular, log 1 = 0 and log 10 = 1. • The functions 10 and log are inverses, so they “undo” each other: log(10 ) = for all , 10 log = for > 0. • For and both positive and any value of , log() = log + log log ( ) = log − log log( ) = ⋅ log . Example 5 Solve 100 ⋅ 2 = 337,000,000 for . Solution Dividing both sides of the equation by 100 gives 2 = 3,370,000. Taking logs of both sides gives log ( 2 ) = log(3,370,000). Since log(2 ) = ⋅ log 2, we have log 2 = log(3,370,000), so, solving for , we have = log(3,370,000) log 2 = 21.684. In Example 2 on page 142, we estimated graphically that the fine faced by the city of Yonkers ex- ceeded the city’s annual budget between day 21 and day 22. The Natural Logarithm When is used as the base for exponential functions, computations are easier with the use of another logarithm function, called log base . The log base is used so frequently that it has its own notation: ln , read as the natural log of . We make the following definition: For > 0, ln is the power of that gives or, in symbols, = ln means = , and is called the natural logarithm of .
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