Section 4.1Logarithms and their Properties

• Suppose you have $100 in an account paying 5% compounded annually.– Create an equation for the balance B after t years– When will the account be worth $200?

• In the previous example we needed to solve for the input

• Since exponential functions are 1-1, they have an inverse

• The inverse of an exponential function is called the common logarithm function or the log function

• In other words

xxy y 10thenlogIf

Example• Simplify the following expressions using logs

16227766.310

00001.010

000,10105

4

• Logarithms are just exponents

• Evaluate the following:

)10log(

)10log(

)000,100log(

10log

1log

• Logarithms are inverses of exponential functions so

• Evaluate

N

N

N

N

N

N

log10

,0forand

)10log(

,anyfor

)1log(222.2log5.7 2

1010)10log( x

Properties of the common Logarithm

btb

bab

a

baab

xx

xx

xxy

t

x

x

y

log)log(

logloglog

loglog)log(

0for10

allfor)10log(

110logand01log

10thenlogIf

log

The Natural Logarithm

btb

bab

a

baab

xxe

xxe

e

xexy

t

x

x

y

ln)ln(

lnlnln

lnln)ln(

0for

allfor)ln(

1lnand01ln

thenlnIf

ln

Evaluate

10

2

1ln

10

100log

)87.0(2210

10010

e

eq

x