7.4a Notes – Evaluate Logarithms. 1. Solve for x. a. x = 2 b. c.d. x = 1 x = 0 x = -2.

7.4a Notes – Evaluate Logarithms

1. Solve for x.

2 4x a.

x = 2

b.

c. d.

3 3x

x = 1

5 1x x = 0

16

36x

x = -2

Logarithms allow us to easily solve for x when it is an exponent.

xb y if and only if logb y x

To find the exponent with e, we have to take the natural log (ln).

xe y if and only if lne y x

These equations mean the same thing. One is written in exponential form and the other is written in log form. When there is no base written for logs, it is assumed the base is 10. Natural logs always have a base of e. Remember to use the “hook” to solve the log form.

2. Rewrite the equation in exponential form.

2log 8 3a.

23 = 8


b.

40 = 1

4log 1 0


c.

121 = 12

12log 12 1


d.1/4log 4 1

11

44


e. 41

log 216

2 14

16

ln 1e e


f.

e1 = e

Properties:

log 1 _____b

log _____b b

ln _____e e

0 because

because

because

b0 = 1

b1 = b1

1 e1 = e

3. Evaluate the logarithm without using a calculator.

a.5log 5 = ?

5? = 5

1


b.= ?

9? = 81

2

9log 81


c.= ?

2? = 32

5

2log 32


d.= ?

–2

1/2log 4

?1

42


e.= ?

–3

? 14

64

41

log64


f.= ?

1

ln e

e? = e

log100


g.= ?

2

10? = 100

5log 25


h.= ?

undefined

5? = -25

114log 11


i.= ?

4

4(11? = 11)

4(1)


j.= ?

-10

-5(3? = 9)

-5(2)

35log 9


k.= ?

134log 4

? 134 4

13

192ln e


j.= ?

? 192 e e

38

2(19)

4. Use a calculator to evaluate the logarithm. Round answers to three decimal places.

a. log 9 0.954

b. ln 2.17 0.775

c. log 0.101 –0.996

d. ln 150 5.011

Compound Interest Continuously Compounded Interest

Consider an initial principal P deposited into an account that pays interest at an annual rate r (expressed as a decimal), compounded n times per year. The amount A in the account after t years is given by the equation:

1nt

rA P

n

rtA Pe

When interest is compounded continuously, the amount A in an account after t years, where P is the principle and r is the interest rate, is given by the formula:

A

r

t

n

P

balance in account ($ in bank)

interest rate MUST BE IN DECIMAL FORM!!!

time in years that money is in the account

number of times per year that money is compounded (recalculated)

principal in account (starting $)

5. If $1,000 is deposited in an account at 8% annual rate compounded quarterly, find the balance after:

a) 1 yr

1nt

rA P

n

A =

P =

r =

n =

t =

?

1,000

0.08

4

1

4 10.08

1,000 14

A

41,000 1 0.2A

1,000 1.0824A$1,082.43A

5. If $1,000 is deposited in an account at 8% annual rate compounded quarterly, find the balance after:

b) 50 yrs

1nt

rA P

n

A =

P =

r =

n =

t =

?

1,000

0.08

4

25

4 500.08

1,000 14

A

2001,000 1 0.2A

1,000 52.484897A$52,484.90A

6. If $1,000 is deposited in an account at 8% annual rate compounded monthly, find the balance after:

a) 1 yr

1nt

rA P

n

A =

P =

r =

n =

t =

?

1,000

0.08

12

1

12 10.08

1,000 112

A

121,000 1 0.0067A

1,000 1.083A$1,083.00A

6. If $1,000 is deposited in an account at 8% annual rate compounded monthly, find the balance after:

b) 50 yrs

1nt

rA P

n

A =

P =

r =

n =

t =

?

1,000

0.08

12

50

12 500.08

1,000 112

A

6001,000 1 0.0067A

1,000 53.87818A$53,878.18A

7. If $1000 is deposited in an account that pays 8% annual interest compounded continuously, find the balance after 50 years.

rtA Pe

A =

P =

r =

t =

?

1,000

0.08

50

0.08 501,000A e 41,000A e

1,000 54.59815A

$54,598.15A

8. Would you prefer to have your account compound annually or continuously? Why?

You get more money!

Compounded continuously

7.4a Notes – Evaluate Logarithms. 1. Solve for x. a. x = 2 b. c.d. x = 1 x = 0 x = -2.

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Transcript of 7.4a Notes – Evaluate Logarithms. 1. Solve for x. a. x = 2 b. c.d. x = 1 x = 0 x = -2.