LESSON Properties Of Logarithms 5.7 B - Prek 12
L ESSO N5.7Each problem that I solvedbecame a rule which servedafterwards to solve otherproblems.Properties of LogarithmsBefore machines and electronics were adapted to do multiplication, division, andraising a number to a power, scientists spent long hours doing computations byhand. Early in the 17th century, Scottish mathematician John Napier (1550–1617)discovered a method that greatly reduced the time and difficulty of these calculations,using a table of numbers that he named logarithms. As you learned in Lesson 5.6,a common logarithm is an exponent—the power of 10 that equals a number—andyou already know how to use the multiplication, division, and power properties ofexponents. In the next example you will discover some shortcuts and simplifications.After inventing logarithms,John Napier designed a devicefor calculating with logarithmsin 1617. Later called “Napier’sbones,” the device usedmultiplication tables carved onstrips of wood or bone. Thecalculator at left has an entire setof Napier’s bones carved on eachspindle. You can learn moreabout Napier’s bones and earlycalculating devices atRENÉ DESCARTESflourishkh.com .EXAMPLEConvert numbers to logarithms to solve these problems.a. Multiply 183.47 by 19.628 without using the multiplication key on yourcalculator.b. Divide 183.47 by 19.628 without using the division key on your calculator.c. Evaluate 4.702.8 without the exponentiation key on your calculator. (You mayuse the 10x key.) SolutionYou can do parts a and b by hand. Or you can convert to logarithms and usealternative functions.a. 183.47 10log 183.47 and 19.628 10log 19.628183.47 19.628 10log 183.47 10 log 19.628 10log 183.47 log 19.628 艐 10 3.556441艐 3601.14910 log 183.47 10 log 183.47 log 19.628 10 0.970689 9.34736183.47 b.19.628 10 log 19.628c.4.70 10log 4.704.702.8 (10log 4.70)2.8 102.8 log 4.70 艐 101.8819 艐 76.2People did these calculations with a table of base-10 logarithms before there werecalculators. For example, they looked up log 183.47 and log 19.628 in a table andadded them. Then they worked backward in their table to find the antilog, orantilogarithm, of that sum.LESSON 5.7 Properties of LogarithmsDAA2SE 984 05 3rd10.indd 29329310/27/12 7:49:55 PM
.8831.8887.8943.8998log 4.70 0.6721antilog 0.8819 7.62antilog 1 10antilog 1.8819 7.62 10 76.2Can you see why “10 to the power” came to be called the antilog? The antilog of 3is the same as 10 3, which equals 1000. Later, slide rules were invented to shortenthis process, although logarithm tables were still used for more precise calculations.Because logarithms are exponents, they must have properties similar to theproperties of exponents. In the following investigation you will use your calculatorto discover these properties.InvestigationProperties of LogarithmsStep 1Use your calculator to complete the table.Record the values to three decimal places.Step 2Look closely at the values for the logarithmsin the table. Look for pairs of values that addup to a third value in the table. For example,add log 2 and log 3. Where can you find thatsum in the table?Record the equations that you find in the formlog 2 log 3 log ? . (Hint: You shouldfind
Practice Your Skills 1. Use the properties of logarithms to rewrite each expression as a single logarithm. a. log 5 log 11 a b. 3 log 2 c. log 28 log 7 a d. 2 log 6 e. log 7 2 log 3 2. Rewrite each expression as a sum or difference of logarithms by using the properties of logari
7. On what other properties do the proofs of the properties of logarithms rely? 8. What properties of logarithms would you use to rewrite the expression log 7 x log 7 4 x as a single logarithm? 9. Explain how the properties of logarithms are useful in finding
BJT and FET Frequency Response Characteristics: -Logarithms and Decibels: Logarithms taken to the base 10 are referred to as common logarithms, while logarithms taken to the base e are referred to as natural logarithms. In summary
Chapter 7: Logarithmic Functions Section 7.3 168 Section 7.3: Laws of Logarithms Laws of Logarithms 1. Addition of Logarithms When two logarithms with the same base are added, their arguments are multiplied. logb( ) logb( ) logb( ) 2. Subtraction of Logarithms When two logarith
Properties of Logarithms Substitute 6.9 for M. Multiply both sides by . 3 2 Simplify. Apply the Quotient Property of Logarithms. Repeat process using the data from San Francisco. Apply the Inverse Properties of Logarithms & Exponents. Use a calculator to evaluate. Given the
Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems F-LE.4 Prove simple laws of logarithms F-LE.4.1 Use the definition of logarithms to translate between logarithms in any base F-LE.4.2 Understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their
Logarithms mc-TY-logarithms-2009-1 Logarithms appear in all sorts of calculations in engineering and science, business and economics. Before the days of calculators they were use
4 Step Phonics Quiz Scores Step 1 Step 2 Step 3 Step 4 Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 . Zoo zoo Zoo zoo Yoyo yoyo Yoyo yoyo You you You you
Classical Theory and Modern Bureaucracy by Edward C. Page Classical theories of bureaucracy, of which that of Max Weber is the most impressive example, seem to be out of kilter with contemporary accounts of change within the civil service in particular and modern politico-administrative systems more generally. Hierarchy and rule-bound behaviour seem hard to square with an environment .
