Logarithm Base

11 Key excerpts on "Logarithm Base"

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  eBook - PDF

  Introductory Mathematics

  • 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

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  Precalculus

  A Prelude to Calculus

  • Sheldon Axler(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Similarly, the expression above is not equal to log 2 3, just as f (15) f (5) is usually not equal to f (3). Section 3.1 Logarithms as Inverses of Exponential Functions 207 Logarithms with Any Base We now take up the topic of defining logarithms with bases other than 2. No new ideas are needed for this more general situation—we simply replace 2 by a positive number b 6 = 1. Here is the formal definition. The base b = 1 is excluded because 1 x = 1 for every real number x. Logarithm Suppose b and y are positive numbers, with b 6 = 1. • The Logarithm Base b of y, denoted log b y, is defined to be the number x such that b x = y. Short version: • log b y = x means b x = y. Example 4 (a) Evaluate log 10 1000. (b) Evaluate log 7 49. (c) Evaluate log 3 1 81 . (d) Evaluate log 5 √ 5. solution (a) log 10 1000 = 3 because 10 3 = 1000. (b) log 7 49 = 2 because 7 2 = 49. (c) log 3 1 81 = -4 because 3 -4 = 1 81 . (d) log 5 √ 5 = 1 2 because 5 1/2 = √ 5. Two important identities follow immediately from the definition. Logs have many uses, and the word “log” has more than one meaning. The logarithm of 1 and the logarithm of the base If b is a positive number with b 6 = 1, then • log b 1 = 0; • log b b = 1. The first identity holds because b 0 = 1; the second holds because b 1 = b. The definition of log b y as the number x such that b x = y has the following consequence. Logarithm as an inverse function Suppose b is a positive number with b 6 = 1 and f is the exponential function defined by f ( x) = b x . Then the inverse function of f is given by the formula f -1 (y) = log b y. Because a function and its inverse interchange domains and ranges, the domain of the function f -1 defined by f -1 (y) = log b y is the set of positive numbers, and If y ≤ 0, then log b y is not defined. the range of this function is the set of real numbers. 208 Chapter 3 Exponential Functions, Logarithms, and e Example 5 Suppose f is the function defined by f ( x) = 3 · 5 x-7 .

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  Intermediate Algebra

  Concepts with Applications

  • Charles P. McKeague(Author)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  1. A common logarithm uses the expression log 10 x, which is equal to log x 10 . 2. When the base of a log is not shown, it is assumed to be 1. 3. When the logarithm of a whole number is approximated as a decimal, the decimal part of the answer is called the characteristic. 4. An antilogarithm is a logarithm with a base of e, denoted by ln x = log e x. 721 doubling time exponential equation change-of-base property KEY WORDS A Solve exponential equations. B Use the change-of-base property to calculate logarithms. C Solve application problems involving logarithmic or exponential equations. OBJECTIVES 8.6 Exponential Equations, Change of Base, and Applications 8.6 Exponential Equations, Change of Base, and Applications For items involved in exponential growth, the time it takes for a quantity to double is called the doubling time. For example, if you invest $5,000 in an account that pays 5% annual interest, compounded quarterly, you may want to know how long it will take for your money to double in value. You can find this doubling time if you can solve the equation 10,000 = 5,000 (1.0125) 4t You will see as you progress through this section that logarithms are the key to solving equations of this type. A Exponential Equations Logarithms are very important in solving equations in which the variable appears as an exponent. The equation 5 x = 12 is an example of one such equation. Equations of this form are called exponential equations. Because the quantities 5 x and 12 are equal, so are their common logarithms. We begin our solution by taking the logarithm of both sides. log 5 x = log 12 We now apply the power property for logarithms, log x r = r log x, to turn x from an exponent into a coefficient. x log 5 = log 12 Dividing both sides by log 5 gives us x = log 12 _____ log 5 If we want a decimal approximation to the solution, we can find log 12 and log 5 on a calculator and divide.

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  eBook - ePub

  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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  Algebra and Trigonometry

  • Cynthia Y. Young(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  We call this exponent a logarithm (or “log” for short). Words Math x = b y is equivalent to y = log b x. y = log b x Let y = f −1 (x). f −1 (x) = log b x Logarithmic Function For x > 0, b > 0, and b ≠ 1, the logarithmic function with base b is denoted f (x) = log b x, where y = log b x if and only if x = b y We read log b x as “log base b of x.” 5.2 Logarithmic Functions and Their Graphs 441 STUDY TIP log b x = y is equivalent to b y = x. This definition says that x = b y (exponential form) and y = log b x (logarithmic form) are equivalent. One way to remember this relationship is by adding arrows to the logarithmic form: log b x = y ⇔ b y = x EXAMPLE 1 Changing from Logarithmic Form to Exponential Form Write each equation in its equivalent exponential form. a. log 2 8 = 3 b. log 9 3 = 1 _ 2 c. log 5( 1 __ 25 ) = −2 Solution a. log 2 8 = 3 is equivalent to 2 3 = 8 b. log 9 3 = 1 _ 2 is equivalent to 9 1/2 = 3 c. log 5( 1 __ 25 ) = −2 is equivalent to 5 −2 = 1 __ 25 Your Turn Write each equation in its equivalent exponential form. a. log 3 9 = 2 b. log 16 4 = 1 _ 2 c. log 2( 1 _ 8 ) = −3 Answer a. 3 2 = 9 b. 16 1/2 = 4 c. 2 −3 = 1 _ 8 EXAMPLE 2 Changing from Exponential Form to Logarithmic Form Write each equation in its equivalent logarithmic form. a. 16 = 2 4 b. 9 = √ _ 81 c. 1 _ 9 = 3 −2 d. x a = z Solution a. 16 = 2 4 is equivalent to log 2 16 = 4 b. 9 = √ _ 81 = 81 1/2 is equivalent to log 81 9 = 1 _ 2 c. 1 _ 9 = 3 −2 is equivalent to log 3( 1 _ 9 ) = −2 d. x a = z is equivalent to log x z = a Your Turn Write each equation in its equivalent logarithmic form. a. 81 = 9 2 b. 12 = √ _ 144 c. 1 __ 49 = 7 −2 d. y b = w Answer a. log 9 81 = 2 b. log 144 12 = 1 _ 2 c. log 7 ( 1 _ 49 ) = −2 d. log y w = b 442 CHAPTER 5 Exponential and Logarithmic Functions Some logarithms can be found exactly, whereas others must be approximated. Example 3 illustrates how to find the exact value of a logarithm. Example 4 illustrates approximating values of logarithms with a calculator.

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  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  You’ve seen these before, no doubt, but here they are again to remind you. For any base b > 0 and real numbers x and y : 1. b 0 = 1 . The zeroth power of any nonzero number is 1. 2. b 1 = b. The first power of a number is just the number itself. 3. b x b y = b x + y . When you multiply two exponentials with the same base, you add the exponents. 4. b x b y = b x -y . When you divide two exponentials with the same base, you subtract the bottom exponent from the top one. 5. ( b x ) y = b xy . When you take the exponential of the exponential, you multiply the exponents. You should also know what the graphs of exponentials look like. We looked at this a little in Section 1.6 in Chapter 1, but in any case we’ll revisit the graph shortly. 9.1.2 Review of logarithms Logarithms—a word that strikes fear into the hearts of many students. Watch carefully, and we’ll see how to deal with these beasts. Suppose that you want to solve the following equation for x : 2 x = 7 . The way you can bring x down from the exponent is to hit both sides with a logarithm. Since the base on the left-hand side is 2, the base of the logarithm is 2. Indeed, by definition, the solution of the above equation is x = log 2 (7) . In other words, to what power do you have to raise 2 in order to get 7? The answer is log 2 (7). This particular number can’t be simplified, but how about log 2 (8)? Ask yourself, to what power do you raise the base 2 in order to get 8? Since 2 3 = 8, the power we need is 3. So log 2 (8) = 3. Let’s go back to the equation 2 x = 7. We know that this means that x = log 2 (7). If we now plug that value of x into the original equation, we get the bizarre looking formula 2 log 2 (7) = 7 . In more generality, log b ( y ) is the power you have to raise the base b to in order to get y . This means that x = log b ( y ) is the solution of the equation b x = y for given b and y .

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  Algebra

  Form and Function

  • William G. McCallum, Eric Connally, Deborah Hughes-Hallett(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Even though we get the right numerical answer in both cases, our solution to Example 1 involves an extra factor of 1∕ log . This is because we used log base 10 to solve an equation involving base . Example 4 Find when the investment from Example 3 will be worth $5000 using the: (a) log function (b) ln function. Solution We have: (a) 1000(1.082)  = 5000 (b) 1000(1.082)  = 5000 1.082  = 5 1.082  = 5 dividing both sides by 1000 log ( 1.082  ) = log 5 ln ( 1.082  ) = ln 5 applying (a) log, (b) ln  log 1.082 = log 5  ln 1.082 = ln 5  = log 5 log 1.082  = ln 5 ln 1.082 = 20.421. = 20.421. 278 Chapter 7 LOGARITHMS In Example 4, we obtain the same numerical answer using either log or ln. In summary, In all cases, you can get the right numerical answer using either ln or log. However: • If an equation involves base 10, using log might be easier than ln. • If an equation involves base , using ln might be easier than log. Otherwise, there is often no strong reason to prefer one approach to the other. 9 Logarithms to Other Bases In addition to base 10 and base , other bases are sometimes used for logarithms. Perhaps the most frequent of these is log 2 , the logarithm with base 2. For instance, since computers represent data internally using the base 2 number system, computer scientists often use logarithms to base 2. To see how these different bases work, suppose we want to solve the equation   =  where  and  are positive numbers. Taking the logarithm of both sides gives log   = log   ⋅ log  = log   = log  log  . This means  is the exponent of  that yields , and we say that  is the “log base ” of , written  = log  . As you can check for yourself, a similar derivation works using ln  instead of log . By analogy with common and natural logarithms, we have: If  and  are positive numbers, log   is the exponent to which we raise  to get . In other words, if  = log   then   =  and if   =  then  = log  .

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  College Algebra

  • Sheldon Axler(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  If your calculator has a button labeled “log”, then it will compute the Logarithm Base 10, which is often just called the logarithm. section 5.2 Logarithms as Inverses of Exponential Functions 297 Common Logarithms and the Number of Digits Note that 10 1 is a two-digit number, 10 2 is a three-digit number, 10 3 is a four-digit number, and so on. In general, 10 n-1 is an n-digit number. Because 10 n , which consists of 1 followed by n zeros, is the smallest positive integer with n + 1 digits, we see that every integer in the interval [10 n-1 , 10 n ) has n digits. Because log 10 n-1 = n - 1 and log 10 n = n, this implies that an n-digit positive integer has a logarithm in the interval [n - 1, n). Alternative statement: If M is a positive inte- ger with n digits, then n - 1 ≤ log M < n. Digits and logarithms The logarithm of an n-digit positive integer is in the interval [n - 1, n). The conclusion above is often useful in making estimates. For example, without using a calculator we can see that the number 123456789, which has nine digits, has a logarithm between 8 and 9 (the actual value is about 8.09). The next example shows how to use the conclusion above to determine the number of digits in a number from its logarithm. example 3 Suppose M is a positive integer such that log M ≈ 73.1. How many digits does M have? solution Because 73.1 is in the interval [73, 74), we can conclude that M is a Always round up the logarithm of a num- ber to determine the number of digits. Here log M ≈ 73.1 is rounded up to show that M has 74 digits. 74-digit number. Logarithm of a Power We will use the formula (b r ) t = b tr to derive a formula for the logarithm of a power. First we look at an example. To motivate the formula for the logarithm of a power, we note that An expression such as log 10 12 should be interpreted to mean log(10 12 ), not (log 10) 12 . log (10 3 ) 4 = log 10 12 = 12 and log 10 3 = 3. Putting these equations together, we see that log (10 3 ) 4 = 4 log 10 3 . More generally, logarithms convert powers to products, as we will now show. Suppose b and y are positive numbers, with b 6= 1, and t is a real number. Then log b y t = log b (b log b y ) t = log b b t log b y = t log b y.

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  Intermediate Algebra

  Concepts and Graphs 2E

  • Charles P. McKeague(Author)
  • 2019(Publication Date)
  • XYZ Textbooks

    (Publisher)

  B Calculate logarithms using the change-of-base property. C Solve application problems involving logarithms or exponential equations. © istockphoto.com/gemenacom 8.6 VIDEOS 8.6 Exponential Equations and Change of Base 703 The complete problem looks like this: 5 x = 12 log 5 x = log 12 x log 5 = log 12 x = log 12 _____ log 5 ≈ 1.0792 _____ 0.6990 ≈ 1.5439 Here is another example of solving an exponential equation using logarithms. EXAMPLE 1 Solve for x: 25 2x+1 = 15 SOLUTION Taking the logarithm of both sides and then writing the exponent (2x + 1) as a coefficient, we proceed as follows: 25 2x+1 = 15 log 25 2x+1 = log 15 Take the log of both sides. (2x + 1)log 25 = log 15 Property 3 2x + 1 = log 15 _____ log 25 Divide by log 25. 2x = log 15 _____ log 25 − 1 Add −1 to both sides. x = 1 __ 2  log 15 _____ log 25 − 1  Multiply both sides by 1 __ 2 . Using a calculator, we can write a decimal approximation to the answer: x ≈ 1 __ 2  1.1761 _____ 1.3979 − 1  ≈ 1 __ 2 (0.8413 − 1) ≈ 1 __ 2 (−0.1587) ≈ −0.079 If you invest P dollars in an account with an annual interest rate r that is compounded n times a year, then t years later the amount of money in that account will be A = P  1 + r _ n  nt EXAMPLE 2 How long does it take for $5,000 to double if it is deposited in an account that yields 5% interest compounded once a year? 704 CHAPTER 8 Exponential and Logarithmic Functions SOLUTION Substituting P = 5,000, r = 0.05, n = 1, and A = 10,000 into our formula, we have 10,000 = 5,000(1 + 0.05) t 10,000 = 5,000(1.05) t 2 = (1.05) t Divide by 5,000. This is an exponential equation. We solve by taking the logarithm of both sides: log 2 = log(1.05) t = t log 1.05 Dividing both sides by log 1.05, we have t = log 2 ______ log 1.05 ≈ 14.2 It takes a little over 14 years for $5,000 to double if it earns 5% interest per year, compounded once a year. Change of Base There is a fourth property of logarithms we have not yet considered.

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  Precalculus: Mathematics for Calculus, International Metric Edition

  • James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 358 CHAPTER 4 ■ Exponential and Logarithmic Functions EXAMPLE 5 ■ Evaluating Logarithms with the Change of Base Formula Use the Change of Base Formula and common or natural logarithms to evaluate each logarithm, rounded to five decimal places. (a) log 8 5 (b) log 9 20 SOLUTION (a) We use the Change of Base Formula with b  8 and a  10: log 8 5  log 10 5 log 10 8 < 0.77398 (b) We use the Change of Base Formula with b  9 and a  e: log 9 20  ln 20 ln 9 < 1.36342 Now Try Exercises 59 and 61 ■ EXAMPLE 6 ■ Using the Change of Base Formula to Graph a Logarithmic Function Use a graphing calculator to graph f 1 x 2  log 6 x . SOLUTION Calculators don’t have a key for log 6 , so we use the Change of Base For- mula to write f 1 x 2  log 6 x  ln x ln 6 Since calculators do have an LN key, we can enter this new form of the function and graph it. The graph is shown in Figure 1. Now Try Exercise 67 ■ CONCEPTS 1. The logarithm of a product of two numbers is the same as the of the logarithms of these numbers. So log 5 1 25 # 125 2   . 2. The logarithm of a quotient of two numbers is the same as the of the logarithms of these numbers. So log 5 A 25 125 B   . 3. The logarithm of a number raised to a power is the same as the times the logarithm of the number. So log 5 1 25 10 2  # . 4. We can expand log a x 2 y z b to get . 5. We can combine 2 log x  log y  log z to get . 6. (a) Most calculators can find logarithms with base and base . To find logarithms with different bases, we use the Formula. To find log 7 12, we write log 7 12  log log < (b) Do we get the same answer if we perform the calculation in part (a) using ln in place of log? 7–8 ■ True or False? 7. (a) log1 A  B 2 is the same as log A  log B. (b) log AB is the same as log A  log B. 8. (a) log A B is the same as log A  log B. (b) log A log B is the same as log A  log B.

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  Explorations in College Algebra

  • Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  To solve this equation, we need natural logarithms that use e as a base. The Natural Logarithm The common logarithm uses 10 as a base. The natural logarithm uses e as a base and is written ln x rather than log e x. Scientific calculators have a key that computes ln x. The Natural Logarithm The Logarithm Base e of x is the exponent of e needed to produce x. Logarithms base e are called natural logarithms and are written as ln x. ln x = c means that e c = x (x > 0) The properties for natural logarithms (base e) are similar to the properties for common loga- rithms (base 10). Like the common logarithm, ln A is not defined when A ≤ 0. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e 345 Assume A and B are positive real numbers and p is any real number. Rules of Common Logarithms 1. log(A ⋅ B) = log A + log B 3. log A p = p log A 2. log(A/B) = log A − log B 4. log 1 = 0 (since 10 0 = 1) Rules of Natural Logarithms 1. ln(A ⋅ B) = ln A + ln B 3. ln A p = p ln A 2. ln(A/B) = ln A − ln B 4. ln 1 = 0 (since e 0 = 1) We can use the rules of natural logarithms to manipulate expressions involving natural logs. EXAMPLE 1 Expanding Expressions with Natural Logs Expand, using the laws of logarithms, the expression: x x ln 3 2 + − . Solution Rewrite using exponents x x x x ln 3 2 ln 3 2 1/2 + − = + −       Rule 3 of ln x x 1 2 ln 3 2 = + −       Rule 2 of ln x x 1 2 [ln( 3) ln( 2)] = + − − EXAMPLE 2 Contracting Expressions with Natural Logs Contract, expressing the answer as a single logarithm: x x ln( 1 ) ln( 1) 1 3 1 3 − + + .

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