The Inverse Function of an Exponential Function Is the Logarithmic Function

Logarithmic Functions

f x  logb x  The inverse function of an exponential function is the ______.  For all positive real numbers x and b, b>0 and b  1, ______if and only if______.  The domain of the logarithmic function is ______ The range of the function is ______

 Since the log function is the inverse of the exponential function, their graphs are symmetric with respect to ______.

** Remember that since the log function is the inverse of the exponential function, we can simply swap ______of our important points. y=x x x y b y logb x y  2 __x__y__ __x__y_ -1 1/b 0 1 1 b

y log x y log2 x 2 __x___y__ -1 0 1

•Logarithmic Functions where b  1 are ______, ______functions. •Logarithmic Functions where 0b  1 are ______, ______functions. •The parent form of the graph has an x-intercept at _____ and passes through ______and ______. •There is a vertical asymptote at ______. •The value of _____ determines the ______of the curve. •The function is neither ______nor ______. There is no ______. •There is no ______. More Characteristics of f x  logb x .

•The domain is ______. • The range is ______• End Behavior: ______• The x-intercept is ______• The vertical asymptote is ______• There is ______y-intercept. • There are ______horizontal asymptotes. • This is a ______, ______function. • It is concave ______.

Domain:______

 Graph f x  log3 x . Range: ______y log3 x x-int.: ______ 3 y  x  V. Asy.:______Inc./Dec.?______Concavity? ______ Important Points: End Behavior: ______

 Graph y log 1 x Domain:______2 Range: ______x-int.: ______V. Asy.:______Inc./Dec.?______Concavity? ______End Behavior: ______

TRANSFORMATIONS

y log x 1 y2log 1 x  3 ylog x 1  2 2 2 3   ______

Domain:______Domain:______Domain:______Range: ______Range: ______Range: ______x-int.: ______x-int.: ______x-int.: ______V. Asy.:______V. Asy.:______V. Asy.:______Inc./Dec.?______Inc./Dec.?______Inc./Dec.?______Concavity? ______Concavity? ______Concavity? ______End Behavior: ______End Behavior: ______End Behavior: ______

More Transformations

h x logb  ax  c

The asymptote of a logarithmic function of this form is the line______. To find an x-intercept in the form, let y=0 in the equation______. To find a y-intercept in this form, let x=0 in the equation______.

Try with ylog3 2 x  1 ylog3 2 x  1 ______

Domain:______Range: ______x-int.: ______V. Asy.:______Inc./Dec.?______Concavity? ______End Behavior: ______

Transformations – Common Log Graph: h x log3 x  10 . ______Domain:______Range: ______x-int.: ______V. Asy.:______Inc./Dec.?______Concavity? ______End Behavior: ______Transformations – Natural Log

Graph: h x  2ln 4 x  8  2 ______

Domain:______Range: ______x-int.: ______V. Asy.:______Inc./Dec.?______Concavity? ______End Behavior: ______

Change of Base Formula log x log x  b logb

Use this formula for entering logs with bases other than 10 or e in your graphing calculator.

If you wanted to graph ylog3  x 1 , you would enter ______in your calculator. Either the ______or the ______log may be used in the change of base formula. So you could also enter______in your calculator.