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log. log. log. log. a. . 8. 3. 2 3. 8. =. =. 4. 12. 2. 1/4. b. . 1. 0. 4 0. 1. =. =. c. . 12. 1. 12 1. 12. =. =. –1. 1. d. . 4. –1. 4. =. =. 4. EXAMPLE 1. Rewrite logarithmic equations. Logarithmic Form. Exponential Form. log. log. log. log.

Transcripts

log a. 8 3 2 3 8 = 4 12 2 1/4 b. 1 0 4 0 1 = c. 12 1 12 1 12 = –1 1 d. 4 –1 4 = 4 EXAMPLE 1 Rewrite logarithmic conditions Logarithmic Form Exponential Form

log 3 7 1/2 14 1. 81 4 3 4 81 = 2. 7 1 7 1 7 = –5 1 3. 1 0 14 0 1 = 2 4. 32 –5 32 = for Example 1 GUIDED PRACTICE Rewrite the condition in exponential shape. Logarithmic Form Exponential Form

log a. b. 25 x 10 log 4 5 4 a. 10 log 4 = b log x = x b ( ) log 25 x b. = 5 2 x 5 2 x = 2 x = b x log = x b EXAMPLE 5 Use opposite properties Simplify the expression. Arrangement Express 25 as a power with base 5 . Force of a power property

y = ln ( x + 3) log a. b. y = 6 x 6 a. From the meaning of logarithm, the backwards of is x . y = 6 y = x e x = ( y + 3) e x – 3 y = ANSWER The opposite of y = ln ( x + 3) is y = e x – 3 . Illustration 6 Find backwards works Find the opposite of the capacity. Arrangement b. y = ln ( x + 3) Write unique capacity. x = ln ( y + 3) Switch x and y . Write in exponential frame. Explain for y .

log 10. 8 log x 8 7 8 x = b = x log x log b 8 b 7 –3 x 11. = x log a x –3 x 7 –3 x = a for Examples 5 and 6 GUIDED PRACTICE Simplify the expression. Arrangement SOLUTION

log 12. 64 x 2 log 64 x = 2 6 x = ( ) 2 6 x 6 x = b x log = x b 13. e ln 20 e 20 = x log x log 20 e for Examples 5 and 6 GUIDED PRACTICE Simplify the expression. Arrangement Express 64 as a power with base 2 . Force of a power property SOLUTION e ln 20

14. Locate the opposite of y = 4 x log 4 From the meaning of logarithm, the reverse of y = x . y = 6 is 15. Locate the opposite of y = ln ( x – 5) . e x = ( y – 5 ) e x + 5 y = The backwards of y = ln ( x – 5) is y = e x + 5 . Respond in due order regarding Examples 5 and 6 GUIDED PRACTICE SOLUTION y = ln ( x – 5) Write unique capacity. x = ln ( y – 5 ) Switch x and y . Write in exponential frame. Understand for y .

log a. y x = 3 EXAMPLE 7 Graph logarithmic capacities Graph the capacity. Arrangement Plot a few helpful focuses, for example, (1, 0) , (3, 1) , and (9, 2) . The y - pivot is a vertical asymptote. From left to right , draw a bend that begins just to one side of the y - hub and climbs through the plotted focuses, as demonstrated as follows.

log b. y x = 1/2 EXAMPLE 7 Graph logarithmic capacities Graph the capacity. Arrangement Plot a few advantageous focuses, for example, (1, 0) , (2, –1) , (4, –2) , and (8, –3) . The y - pivot is a vertical asymptote. From left to right , draw a bend that begins just to one side of the y - pivot and moves down through the plotted focuses, as demonstrated as follows.

Graph . Express the space and range. y ( x + 3) + 1 = log 2 Sketch the diagram of the parent work y = x , which goes through (1, 0), (2, 1) , and (4, 2) . Case 8 Translate a logarithmic diagram SOLUTION STEP 1 STEP 2 Translate the parent chart left 3 units and up 1 unit. The deciphered chart goes through (–2, 1) , (–1, 2) , and (1, 3) . The diagram\'s asymptote is x = –3 . The space is x > –3 , and the range is all genuine numbers.

log 16. y x = 5 If x = 1 y = 0 , x = 5 y = 1 , x = 10 y = 2 Plot a few helpful focuses, for example, (1, 0) , (5, 1) , and (10, 2) . The y - hub is a vertical asymptote. for Examples 7 and 8 GUIDED PRACTICE Graph the capacity. Express the area and range. Arrangement

for Examples 7 and 8 GUIDED PRACTICE From left to right , draw a bend that begins just to one side of the y - pivot and climbs through the plotted focuses. The space is x > 0 , and the range is all genuine numbers.

log 17. y ( x – 3) = 1/3 for Examples 7 and 8 GUIDED PRACTICE Graph the capacity. Express the area and range. Arrangement space: x > 3 , extend: every single genuine number

log 18. y ( x + 1) – 2 = 4 for Examples 7 and 8 GUIDED PRACTICE Graph the capacity. Express the space and range. Arrangement area: x > 21 , run: every single genuine number

log b. 0.2 a. 64 b 5 4 To help you discover the estimation of y , solicit yourself what control from b gives you y . a. 4 to what control gives 64 ? 4 3 64 , so 64 3 . = b. 5 to what control gives 0.2 ? 0.2 , so 0.2 –1 . 5 –1 log = 5 EXAMPLE 2 Evaluate logarithms Evaluate the logarithm. Arrangement

log d. 125 c. 6 1/5 36 1/5 b 36 –3 1 To help you discover the estimation of y , solicit yourself what control from b gives you y . 5 c. what exactly control gives 125 ? 125 , so 125 –3 . = 1 36 1/2 6 , so 6 . d. 36 to what control gives 6 ? = 2 EXAMPLE 2 Evaluate logarithms Evaluate the logarithm. Arrangement

log 8 a. 8 10 0.903 8 b. ln 0.3 .3 e –1.204 0.3 EXAMPLE 3 Evaluate regular and characteristic logarithms Expression Keystrokes Display Check 0.903089987 – 1.203972804

Tornadoes The wind speed s (in miles every hour) close to the focal point of a tornado can be demonstrated by s 93 log d + 65 = where d is the separation (in miles) that the tornado voyages. In 1925 , a tornado voyaged 220 miles through three states. Gauge the twist speed close to the tornado\'s inside. Case 4 Evaluate a logarithmic model

s 93 log d + 65 = 93(2.342) + 65 ANSWER The twist speed close to the tornado\'s middle was around 283 miles for every hour. Case 4 Evaluate a logarithmic model SOLUTION Write work. = 93 log 220 + 65 Substitute 220 for d . Utilize a mini-computer. = 282.806 Simplify.

log 5. 32 27 2 5 32 , so 32 5 . = 6. 3 27 1/3 , so 3 . 1 = 3 for Examples 2, 3 and 4 GUIDED PRACTICE Evaluate the logarithm. Utilize a number cruncher if essential. Arrangement 2 to what control gives 32 ? Arrangement 27 to what control gives 3 ?

log 12 7. 12 10 1.079 12 8. ln 0.75 .75 e –0.288 0.75 for Examples 2, 3 and 4 GUIDED PRACTICE Evaluate the logarithm. Utilize a number cruncher if fundamental. Expression Keystrokes Display Check 1.079 – 0.288

9. WHAT IF? Utilize the capacity in Example 4 to gauge the twist speed almost a tornado\'s middle if its way is 150 miles in length. s 93 log d + 65 = 93(2.1760) + 65 ANSWER The twist speed close to the tornado\'s middle is around 267 miles for every hour. for Examples 2, 3 and 4 GUIDED PRACTICE SOLUTION Write work. = 93 log 150 + 65 Substitute 150 for d . Utilize an adding machine. = 267 Simplify.

Use 3 0.792 and 7 1.404 to assess the logarithm. log 4 – 7 a. 3 = and log 3 Use the given estimations of Use the given estimations of – 0.792 1.404 log 7 . 7 . 4 3 –0.612 = 7 Write 21 as 3 7 . b. (3 7) 21 = 3 7 = + 0.792 1.404 2.196 = EXAMPLE 1 Use properties of logarithms Quotient property Simplify. Item property Simplify.

Use 3 0.792 and 7 1.404 to assess the logarithm. log 4 c. 49 7 2 = Write 49 as 7 2 7 = log 7 . 4 2(1.404) Use the given estimation of 2.808 = EXAMPLE 1 Use properties of logarithms Power property Simplify.

Use 5 0.898 and 8 1.161 to assess the log 6 logarithm. log 6 – 8 log 1. 5 = 6 and Use the given estimations of and log 5 Use the given estimations of – 0.898 1.161 log 8 . 8 . 6 5 –0.263 = 8 Write 40 as 8 5 . 2. (8 5) 40 = 8 5 = + 1.161 0.898 2.059 = for Example 1 GUIDED PRACTICE Quotient property Simplify. Item property Simplify.

Use 5 0.898 and 8 1.161 to assess the log 6 logarithm. log 6 3. 64 8 2 = Write 64 as 8 2 8 = log 8 . 5 . 6 2(1.161) Use the given estimation of Use the given estimation of 2.322 = 4. 125 5 3 = Write 125 as 5 3 5 = 3(0.898) 2.694 = for Example 1 GUIDED PRACTICE Power property Simplify. Control property Simplify.

5 x 3 5 x 3 Expand y log 6 – 5 x 3 y = – 5 x 3 + y = – 3 5 x + y = EXAMPLE 2 Expand a logarithmic expression SOLUTION Quotient property Product property Power property

– log 9 + 3 log 2 log 3 log 9 + log 2 3 log 3 = – log 3 = (9 ) 2 3 log = 9 2 3 log 24 = ANSWER The right answer is D. Illustration 3 Standardized Test Practice SOLUTION Power property Product property Quotient property Simplify.

5. Grow log 3 x 4 . log 3 x 4 log 3 + log x 4 = log 3 + 4 log x = for Examples 2 and 3 GUIDED PRACTICE SOLUTION Product property Power property

6. Gather ln 4 + 3 ln 3 – ln 12 . – ln 4 + ln 3 ln 12 = – ln 12 = 4 3 ln = (4 ) 3 12 ln 9 = for Examples 2 and 3 GUIDED PRACTICE SOLUTION ln 4 + 3 ln 3 – ln 12 Power property Product property Quotient property Simplify.

Evaluate 8 utilizing regular logarithms and characteristic logarithms. log 3 log 8 0.9031 1.893 = 0.4771 log 3 ln 8 2.0794 1.893 = 1.0986 ln 3 EXAMPLE 4 Use the change-of-base equation SOLUTION Using basic logarithms: Using common logarithms:

Sound Intensity I L ( I ) 10 log = 0 where is the power of a scarcely discernable sound (about watts per square meter). A craftsman in a recording studio turns up the volume of a track so that the sound\'s power copies. By what number of decibels does the uproar increment? 10 –12 EXAMPLE 5 Use properties of logarithms, all things considered, For a sound with force I (in watts per square meter), the commotion L ( I ) of the sound (in decibels) is given by the capacity

L (2 I ) – L