5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 8, 2018 13:46

68 STEP 4. Review the Knowledge You Need to Score High

Properties of Logarithms

Given thatx,y, andaare positive numbers witha=1 andnis a real number, then

loga(xy)=logax+logay

loga

(

x

y

)

=logax−logay

logaxn=nlogax

Note that loga 1 =0, logaa=1, and logaax=x.

The Natural Basee

e≈ 2. 71828182846 ...

The expression

(

1 +

1

x

)x

approaches the number e as x gets larger and larger. An

equivalent expression is (1+h)^1 /h. The expression (1+h)^1 /halso approacheseashapproaches 0.

Exponential Function with Basee:f(x)=ex

The Natural Logarithmic Function:f(x)=lnx=logexwherex>0.

Note thaty=exandy=lnxare inverse functions:elnx=lnex=x. Also note thate^0 =1,

ln 1=0, and lne=1. (See Figure 5.3-26.)

y

x

(0, 1)

(1, 0)

(e, 1)

(1, e)

0

y = ln x

y = ex

y = x

Figure 5.3-26

Properties of the Natural Logarithmic and Exponential Functions

Givenxandyare real numbers, then

ex·ey=ex+y

ex÷ey=ex−y

(ex)y=exy

lnxy=lnx+lny

ln

(

x

y

)

=lnx−lny

lnxn=nlnx