The number e is ________________ It is also known as Euler’s number Irrational

Complete a t-chart Domain: Range: Y-intercept: Asymptote: XY Type of Function: : Growth All reals x Y>0 (0,1) Y=0

Complete a t-chart Domain: Range: Y-intercept: Asymptote: XY Type of Function: : Decay All reals x Y>0 (0,1) Y=0 f(x) =e ax, rep exp growth when a is ___________ and decay when a is _________ Positive negative

SOLVE for x in the following Use the calculator and round 3 places

Change the following exponential fcns to logarithmic form In the next slide you will see why this can also be written as a natural log

If x is a positive real number, then the natural logarithm of x is denoted by: or Note that if a base is not written here – it is base e

Calculators can evaluate logs with the common base….. which is base 10 They can also evaluate the natural logarithm…which is base e, the natural number Use the log key Use the ln key We will do more calculator log problems in 8.5

Inverses These 2 graphs are reflections over the line _________ y = x Exp fcn HA: y=o VA: x=o Natural log fcn Is ________ of inverse

Special Values of Logarithms ln 1 =___ ?=0 ln e =___ln e x =___e ln(x) =___ 0 ?=1X=? 1 X ?=x Hint: HW x e a·ln(x) =___

PRODUCT QUOTIENT POWER

SOLVE for x using the properties of logs ln 12 = ln 3 + ln x

SOLVE for x using the properties of logs 5 ln(x) = 20

SOLVE for x using the properties of logs 5e x =10 e x =2 ln e x = ln 2 x = ln 2 or x = 0.693

Log fcn has VA: no horizontal shift so x=o Can also be written as: This is a log fcn with base e and is to be shifted: The negative causes a reflection in the x axis. The 2 cause a vertical shift up 2 Remember: ln 1 = 0 XY 12

Activity: The following are actually for lesson 8.5 HW : WS 8.4 – which is due next class

X=0.882

X=0.693 Use the change of base formula Or more commonly seen Take ln of both sides To eliminate the base e

X= Take ln of both sides To eliminate the base e

X=3.585 Use the change of base formula Convert to a log Does this equal Log 6

X=3.975

X= Convert to a log Convert to an exponential Or more commonly seen

X=7.389 or

X=18