Section 1.4Logarithmic Functions

Find x for the following:

• How about now?

813

497

24

82

x

x

x

x

102 x

• 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 logarithm function or the log function

• In other words

xxy y 10thenlogIf

• Now our exponential equation may not always have a base of 10. The general definition of log is

• Often we will deal with the natural base which gives way to the natural log

If log then yby x b x

If ln then yy x e 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

Example• The median house price in the Phoenix area in

2000 was about $120,000. In 2005, the median house price rose to about $250,000. Assuming the growth was exponential, create a model for the median house price as a function of time.– Hint: Use t = 0 to correspond to the year 2000.– According to the model, when will the median

house price be $350,000?