Exponential Equations

Like Bases

Warm Up

The following quadratic equation has exactly one solution for x. Find the value of k. Explore more than one method…..

0,0944 2 kkxx

Method One

One solution indicates that the discriminant is equal to 0:

3

0sin3

03316

0916

014416

094416

04

2

2

2

2

k

kcebutk

kk

k

k

k

acb

Method Two

....0944 2 sosquareperfectaiskxx

solutiononeonlyisthereBecause

3

124

91243232

.329442

22

k

k

xxxx

inspectionbyxkxx

Warm Up - If $5,000 is invested at an interest rate of 5% per year, find the amount of the investment at the end of 5 years for the following compounding methods:

Annually

Monthly

Quarterly

Continuously

Annually: Monthly:

41.6381

1

05.015000

1

51

A

A

n

rPA

tn

79.6416

12

05.015000

1

512

A

A

n

rPA

tn

Quarterly: Continuously

19.6410

4

05.015000

1

54

A

A

n

rPA

tn

13.6420

5000 505.0

A

eA

PeA rt

82 23 xSolve

3

1

13

323

22 323

x

x

x

x

429 327 xxSolve

31

42273

33

3342273

4293

x

xx

xx

xx

11

11

104213

33

33104213

52273

x

x

xx

xx

xx

527 927 xxSolve

9275 6432 xxSolve

13

89

8913

54123525

22

2254123525

926755

x

x

xx

xx

xx

82

14

x

Solve

7

7

34

22

2234

341

x

x

x

x

x

279

126

x

Solve

12

7

712

3412

33

333412

3262

x

x

x

x

x

77 3225.0 xxSolve

19

35

3519

35514

22

22

22

1

324

1

35514

35572

757

2

77

x

x

xx

xx

xx

xx

xx

Logarithmic Functions

Logarithms

Used to find unknown exponents in exponential models

Define many measurement scales in the sciences such as the pH, decibel, and Richter scales.

For any positive base b, where b≠1: bx = y if and only if x = logby

Exponential Form

Logarithmic Form

310 1000 10log 1000 3

Write in logarithmic form.

35 125

3125log 5

Write in logarithmic form.

46 1296

41296log 6

Write in logarithmic form. 3 1

327

327

1log 3

Natural log

log lne x x

xe x ln

The Natural Logarithmic Function

The natural logarithmic function

is the inverse of the natural exponential function

)ln(log xydabbreviatexy e

.xey

Write in logarithmic form. 2 7.389...e

2389.7ln

2389.7log

asSame

e

Write in exponential form. 3log 81 4

8134

Write in exponential form. 8log 512 3

51283

Write in exponential form. ln(181.3) 5.2

3.181

2.53.181log

2.5

e

e

Write in exponential form. 3

1log 2

9

9

13 2

Find the value of x.4log 64x

3

44

6443

x

x

x

Find the value of x.2log 16x

4

22

1624

x

x

x

Find the value of x. 125log 5x

3

1

13

55

512513

x

x

x

x

Find the value of x. 5 log 32x

2

232

32

32

5

155

5

15

15

5

x

xorx

x

x

Find the value of x.34 log x

x

x

81

34

Find the value of x. 2log 1x

0

12

x

x

Find the value of x.1

log 92 x

81

9

9

2

2

2

1

2

1

x

x

x

Find the value of x. 73 log x

x

x

343

73

Find the value of x. 9log 3x

4

12

12

33

39

2

12

x

x

x

x

Find the value of x. 1ln xe

1

1

1log

1

x

ee

ee

xe

x

x

e

Solve ( ) ln 44f x x for x

...784.344ln44 f

Find to the nearest thousandth – using the calculator.

10( ) log 240f x x for x

380.2

240log 10

xf

xf

Find to the nearest thousandth – using the calculator.

10 85x

929.1

85log 10

x

x

Find to the nearest thousandth – using the calculator.

10 22x

342.1

22log 10

x

x