Section 8.3 Part #2. Given log 3 5 = x, give log in terms of x. Given log 8 = k, find log 800 in terms of k. Section 8.3 Part#2 BLANK.

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1 Given log 8 = k, find log 800 in terms of k Section 8.3 Part #2 Given log 3 5 = x, give log in terms of x 1

2 If log 5 x = 1.5 and log 5 y = 2.5, evaluate 2

3 If log 3 = P and log 5 = Q then write an algebraic expression for log 45 Answer 2P+Q 3

4 If log 5 7 = K the write an algebraic expression for log Answer 3+K 4

5 8.3 Example 3 Use the Laws of Logarithms to Simplify Expressions Write each expression as a single logarithm in simplest form. State the restrictions on the variable. a) b) log 5 (2x 2) log 5 (x 2 + 2x 3) 1 a)

6 8.3 Example 3: Your Turn Write each expression as a single logarithm in simplest form. State the restrictions on the variable. a) 4 log 3 x (log 3 x + 5 log 3 x) b) log 2 (x 2 9) log 2 (x 2 x 6) 6

7 Textbook p. 401 #8 11 (ignore restrictions for now) Student Workbook Booklet from last class p.281 #9 The next problem is on p.398 in your textbook. DO NOT WORRY ABOUT COPYING THE ANSWER DOWN. Just make note about the shortcut (2 slides from now). 7

8 8.3 Example 4 Solve a Problem Involving a Logarithmic Scale The human ear is sensitive to a large range of sound intensities. Scientists have found that the sensation of loudness can be described using a logarithmic scale. The intensity level, β, in decibels, of a sound is defined as, where I is the intensity of the sound, in watts per square metre (W/m 2 ), and I0 is W/m 2, corresponding to the faintest sound that can be heard by a person of normal hearing. a) Audiologists recommend that people should wear hearing protection if the sound level exceeds 85 db. The sound level of a chainsaw is about 85 db. The maximum volume setting of a portable media player with headphones is about 110 db. How many times as intense as the sound of the chainsaw is the maximum volume setting of the portable media player? b) Sounds that are at most times as intense as a whisper are considered safe, no matter how long or how often you hear them. The sound level of a whisper is about 20 db. What sound level, in decibels, is considered safe no matter how long it lasts? 8

9 8.3 Example 4 Continued Solve a Problem Involving a Logarithmic Scale 1 a) Let the decibel levels of two sounds be and. 2 Then, compare the two intensities Copy this down since it is a great shortcut to use although it will look different with different formulas 9

10 8.3 Example 4 Continued Solve a Problem Involving a Logarithmic Scale 4 Substitute β 2 = 110 and β 1 = 85 into the equation 5 6 The ratio of these two intensities is approximately 316. Hence, the maximum volume level of the portable media player is approximately 316 times as intense as the sound of a chainsaw. 10

11 8.3 Example 4 Continued Solve a Problem Involving a Logarithmic Scale 7 b) The ratio of the intensity of sounds considered safe to the intensity of a whisper is to 1. In the equation, substitute β 1 = 20 and = β 2 20 = 10 log β 2 = 10 log β 2 = 10 log β 2 = 10(5) + 20 β 2 = 70 9 Sounds that are 70 db or less pose no known risk of hearing loss, no matter how long they last. 11

12 8.3 Example 4: Your Turn The ph scale is used to measure the acidity or alkalinity of a solution. The ph of a solution is defined as ph = log [H + ], where [H + ] is the hydrogen ion concentration in moles per litre (mol/l). A neutral solution, such as pure water, has a ph of 7. Solutions with a ph of less than 7 are acidic and solutions with a ph of greater than 7 are basic or alkaline. The closer the ph is to 0, the more acidic the solution is. a) A common ingredient in cola drinks is phosphoric acid, the same ingredient found in many rust removers. A cola drink has a ph of 2.5. Milk has a ph of 6.6. How many times as acidic as milk is a cola drink? b) An apple is 5 times as acidic as a pear. If a pear has a ph of 3.8, then what is the ph of an apple? Answer 12

13 Richter scale is a measurement of the intensity of earthquakes. Each increase of one unit on the Richter scale means a ten fold increase in strength (an increase of two is a hundred fold increase). How many times more powerful is an earthquake that measures 6.1 on the Richter scale, than an earthquake that measures 4.8? Already reduced to shortcut form. M=log A 13

14 In 1933, an earthquake in Japan had a magnitude of 8.9 on the Richter scale. In 1964, an earthquake in Alaska had a magnitude of 8.5 on the Richter scale. How many times stronger was the Japan earthquake than the Alaska earthquake? Round off the answer to the nearest tenth. 14

15 Homework Textbook p #13 17 Student Workbook p #5 6 You can try the remaining slides if you want extra practice. 15

16 On the ph scale, seven is neutral. Each unit above seven represents a ten fold increase in alkalinity, and each unit below seven indicates a tenfold increase in acidity. How many times more acidic is orange juice with a ph of 3.5 than black coffee with a ph of 5.1? 16

17 The decibel scale measures the loudness of sounds. The threshold of hearing is zero. Originally, it was called the Bel scale, and on the Bel scale each unit indicated a ten fold increase in loudness. On the decibel scale, we need to divide by 10 (deci means divide by ten). The sound intensity of an object is W/m 2, what is its decibel level? 17

18 How many times louder is an ipod at 110 db than an object at 85 db (the threshold for hearing damage)? 18

19 center/decibel exposure time guidelines/ 19

20 A chainsaw has a decibel level of 120 db and a dishwasher has a decibel level of 80 db. How many times louder is a chainsaw than a dishwasher? 20