Mastering Logarithmic Functions: Concepts and Practice
Explore logarithmic functions through changing, evaluating, graphing, finding domains, and solving equations. Understand the properties and intricacies of logarithmic functions adeptly.
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Presentation Transcript
11.4A LOGARITHMIC FUNCTIONS SWBAT >Change and evaluate logarithmic and exponential functions. >Graph and find the domain of logarithmic functions. >Solve Log equations.
Logarithmic Function of Base b: Logs are inverses of exponential functions The answer to a logarithm expression is an exponent If b>0 and b 1then Natural log is denoted ? = ??? ? = ??? ??= ? ? = ?? ????? = ? Read log base b of y is x Can t take the log of a negative number Natural log is a log with base e
Change the exponential to a log Change the log to an exponential ?. ?? = ?? ?. ????? = ? Answer Answer 70= 1 log416 = 2 ??? ?? ??? ???? ?? ? = ? 2. ??= ?.? 1 9= 2 4. log3 Answer 3 2=1 Answer log34.6 = ? 9
Find the exact value for each log .without a calculator: 6. log?/???? 5. log?? Answer 8?= 8 1 Answer ?= 125 1 5 -3 ??? ???? ? ?? ? ?? ? 7. log39 Answer 8. ???.001 1 2(?)= 9 3 4 -3
Solve each log equation: Because ex and lnx are inverses then ????= ? ??? ????= ? ??? =1 Power property log???= rlog?? ex 9: log??? ? = ? 43= 5? 2 64 = 5? 2 ? =66 5 Ex 11: ? ??+?= ?? Take the ln of both sides ??? 2?+1= ??13 EX 10: log????= ? X= 3 2? + 1 = ??13 ? =??13 1 2 End day 1
Warm-Up 12/7 : Graph each of the following. Show horizontal asymptote as a dotted line ? ? = 2? 4+ 2 ? ? = ? ? 3
Notes 11.4B 1. Given y = 2x Make a table from -2 to 2 Sketch a graph domain Range End behavior x/y-intercepts Increasing/decreasing
Properties of graphs of logs of ? ? = log??: 1) Domain = positive reals; Range = all reals. 2) x-intercept = (1,0); No y-int. 3) The line x=0 is the vertical asymptote. 4) Its decreasing if 0<a<1; Increasing if a>1. 5) The graph of f contains (1,0), (a,1), (1/a,-1)
Finding the domain of a log function: Make the parenthesis greater than 0. ? ? = log?(? ?) ? ? = ? + ???(??) The parenthesis need to equal #s bigger than 0. {?|? > 5} Set 2x> 0 {?|? > 0}
Graph ? ? = log1/3? X F(x) 1 1/3 3 9 0 1 -1 -2 a is a fraction so graph is decreasing. Why did I pick these inputs? Domain: x>0
Graph, determine the domain, and vertical asymptote: ? ? = log2(? +5) 1 *? ? = log?? +? We are going to graph:? ? = log2? . Then we will shift left 5 and down 1. X 1 2 4 F(x) 0 1 2 Domain: x>-5
GRAPH WITHOUT A CALCULATAOR EX: ? = log4? + 1 2 try on your own ? = ??? + 3 Base is ? so (1,0) (?,1) ??? (1/?, 1) Reflected x-axis up 3 Parent function ? = ??
11.4A p 650 #s 1-23 odd, 25-36 all, 71-89 odd 11.4B p. 650 # s 37-43 odd, 55-68all, 72-86 even
