Warm Up Solve the following without graphing.

644.1 x

322.2 2 x

82

1.3

x

100010.4 x

1. x =32. x =2.53. x =-34. x =35. Cant be done…

43610.5 x

Investigation What is the exponent?

10x = 1 10x = 10 10x = 100 10x = 1000 10x = 10000

The log button on the calculator is log base 10.

Calculate the following Log 1 Log 10 Log 100 Log 1000 Log 10000

What is the exponent? 2x = 1 2x = 4 2x = 8 2x = 16 2x = 32

Graph y = log2x and fill in the table.

x y

1

2

8

16

32

Investigation Compare the exponential equations with base

10 to the log equations. What do you notice? Does the same thing appear to happen with

the base 2 equations? How are the exponentials and logs related?

Logarithims Logs = exponents

Used to solve when the variable is in the exponent.

Logs and Exponentials are inverses (undo) of each other.

Examples: Rewrite each of the following.

xnnbb

x log

1000

110.2322.1 35

31000

1log.2532log.1 102

Natural log and Common Log Common Log

Base 10 The log button Not written

Natural log Base e Growth/decay rate What have you talked about in science?

AA10

loglog is

AA elogln is

1000

1log

1000

1log10

45xe

x

is

xe

45ln

45log

Change of Base To calculate logs with other bases

orb

NNb log

loglog

b

NNb ln

lnlog

2log

64log64log2

642

62log

64log

6

2ln

100ln100log2

1002

643856.62ln

100ln

643856.6

Solving with Logs Rewrite as a logarithm and solve. Plug solution back into the original equation to

check.

78.6

75.610.5

1282.4

120.3

45010.2

302.1

12

2

3

x

x

x

x

x

x

e

e

787.4

120ln.3

067.14

450log.2

907.4

30log.1 2

x

x

x

x

x

x

678.2

1278ln.6

0853.

2675.log.53

12

3128log.4 2

x

x

x

x

x

x

Solving with Logs A bank is offering a CD with 3.25% interest

compounded continuously. How long will it take your investment to double?

The half-life of a radio-active material is 245 years. What is the rate of decay?

rtPeA

yearst

t

e t

328.21

0325.2ln

12 0325.

re24512

1

%283.00283.

2452

1ln

orr

r

Summary Logs = _________ To solve when x is in the exponent,

_____________ logA has a base of ___ and can be rewritten

_________ lnA has a base of ___ and can be rewritten

________ Solve these problems on your post-it note and

turn in.

growth? of rate theiswhat

days,5everydoublesBacteria.38

12.2

124310.1

23

24

x

x

.5%86.131386..33

1.2

453..1

dayseveryorx

x

x

Exponents rewrite as a log

10 log10

e loge