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Logarithms Lesson PART I: Eponential Functions Eponential functions: These are functions where the variable is an eponent. The first tpe of eponential graph occurs when the base has a value greater than : = - - - = = - - - - - - = b b > When the base is greater than, the graph starts low and gets higher from left to right. - - - The second tpe occurs when the base has a value between 0 and. - - - - - - - - - = b 0< b < When the base is between 0 and, the graph starts high and gets lower from left to right. - - - Notice that the point (0,) is common to all untransformed eponential graphs. This is because anthing raised to the power of zero is one! You can use this feature as an anchor point when drawing these graphs. All eponential graphs have a horizontal asmptote. In the above graphs, the asmptote occurs along the -ais. (Equation: = 0) The domain of untransformed eponential graphs is R since the graph goes left & right forever. The range is > 0. (The smbol is NOT used, due to the presence of the asmptote.) Remember all of the above rules are based on untransformed eponential graphs. Once transformations are involved, these points & lines will move. Pre Calculus Math 0S: Eplained! www.math0s.com 07
Logarithms Lesson PART II: Logarithmic Functions Logarithmic functions: A logarithmic function is the inverse of an eponential function. = log b Variable To draw log graphs in our TI-3, ou must tpe in log(variable) / log(base). Base There are two basic tpes of log graphs ou will need to memorize: The first tpe occurs when the base of the logarithm is bigger than. = log = log = log - 3 5-3 5 - Eample: To graph log, ou would tpe in log() log() 3 5 = log b > b When the base is greater than, the graph starts low and gets higher from left to right. - - - The second tpe is when the base of the logarithm is between 0 and. - = log = log = log = logb 0< b < 3 5-3 5-3 5 When the base is between 0 &, the graph starts high and gets lower from left to right. - - - Notice that the point (,0) is common to all untransformed log graphs. This occurs because a log graph is the inverse of an eponential graph. So, if eponential graphs have the point (0,), it follows that log graphs should pass through (,0) log graphs have a vertical asmptote. In the above graphs, the asmptote occurs along the -ais. (Equation: = 0) The domain of untransformed log graphs is > 0 since the graph is alwas to the right of the vertical asmptote. The range is R since the graph goes up & down forever. If ou ever have negative numbers, 0, or as a base, no graph eists since the logarithm is undefined. Remember that transformations will change the above values. Pre Calculus Math 0S: Eplained! www.math0s.com 0
Logarithms Lesson I PART II: Logarithmic Functions 5 Eample : Given = Grap h in our calculator as ( 5 ) ^ : [-,, ] : [-,, ] b) What is the domain & range The domain is ε R The range is > 0 The asmptote is the -ais, so the equation is = 0 - - - d) What are the & intercepts? Ther e is no -intercept due to the asmptote.. Find the -intercept b using nd Trace Value = 0 in our TI-3. Answer = (0, ) e) What is the value of the graph when =? You could plug = into the equation and solve, but an easier wa is to use the TI-3. Go nd Trace Value =. This will give ou the resulting -value automaticall. Answer =.5 Eample : Given =log Graph in our calculator as log() log() : [0,, ] : [-,, ] b) Wha t is the d omain & range Th e domain is > 0 due to the vertic al asmptote at the -ais. The range is ε R The asmptote is the -ais, so = 0 d) What are the & intercepts? a logarithm without a base, such as = log, can be tped in as is The intercept can be found b going and ou ll get the proper graph. nd Trace Zero in our TI-3. Answer = (, 0) Ther e is no -intercept du e to the vertical asmptote common logarithms. the actuall have a base of 0, It s just not at the -ais. e) What is the value of the graph when =? logarithm. Go nd Trace Value = Answer = 0.5 3 - - -3 - - - You will alwas have to tpe logarithms into our calculator as a fraction with one eception: logarithms without bases are called written in. The log button on our calculator is a common Pre Calculus Math 0S: Eplained! www.math0s.com 09
Function Graph Domain Range Equation of Asmptote -intercept -intercept -value when = = 3 - - - - - - 3 - - - - - - = log. 3 - - - 3 - - - - - - ( ) = log 3 = + ( ) = log + Pre Calculus Math 0S: Eplained! www.math0s.com 0
Equation of Asmptote Function Graph Domain Range -intercept -intercept - - - - - - -value when = = 3 ε R > 0 = 0 None 9 3 log - - - - - - - - - ε R > 0 = 0 None =. > 0 ε R = 0 None 3.0 3 3 Notice how the base is greater than. - - - - - - ε R > 0 = 0 None.5 9 ( ) = log 3 > 3 ε R = 3 None Undefined = + ε R > = None 3 ( ) = log + > - ε R = - - 0.30 0.0 ε R > - = - - -3-3.75 Pre Calculus Math 0S: Eplained! www.math0s.com
Logarithms Lesson I PART II: Natural Logarithms Natural Logarithms: Logarithms that have a base of.7 are called Natural Logarithms. (.7 is known as Euler s Number, and is represented b the letter e) Eample : Given =e Graph in our calculator using: nd e right above the ln button ( ) Should look like Y = e^() : [-5, 5, ] : [-5, 5, ] b) What is the domain & range The domain is ε R The range is > 0 The asmptote is the -ais, so the equation is = 0 d) What are the & intercepts? There is no -intercept due to the asmptote.. Find the -intercept b using nd Trace Value = 0 in our TI-3. Answer = (0, ) e) What is the value of the graph when =? You could plug = into the equation and solve, but an easier wa is to use the TI-3. Go nd Trace Value =. This will give ou the resulting -value automaticall. Answer = 7.39 Eample : Given =l n( ) Graph in our calculator as ln() : [-5, 5, ] : [-5, 5, ] b) What is the domain & range The domain is > 0 due to the vertic al asmptote at the -ais. The range is ε R The asmptote is the -ais, so = 0 d) What are the & intercepts? The intercept can be found b going nd Trace Zero in our TI-3. Answer = (, 0) There is no -intercept due to the vertical asmptote at the -ais. e) What is the value of the graph when =? Go nd Trace Value = Answer = 0.93 ln() is an alternative wa of epressing log e, and is pronounced Lon = ln() and = e are inverses of each other, and are reflected across the line =. Pre Calculus Math 0S: Eplained! www.math0s.com