CLASS NOTES: §10 – 4 thru §10 – 8mszaleski.weebly.com/.../notes_10_4_thru_10_8.pdf · Algebra...

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CLASS NOTES: §10 – 4 thru §10 – 8 Logarithms §10 – 4: Definition of Logarithms

Logarithmic Form Exponential Form of the of the logarithmic function logarithmic function EX 1 Rewrite each equation in exponential form. (a)

€

2 = log 6 36 (b)

€

log2 2 = 1 (c)

€

log10 0.001( ) = −3

Review of Functions that we know… Linear Function Quadratic Function

Exponential Function

NEW! Logarithmic Function

b is a number

Definition of Logarithm


Logarithmic Form Exponential Form of the of the logarithmic function logarithmic function EX 2 Rewrite each equation in logarithmic form.

(a)

€

16 = 2 4

(b)

€

8 −2 3 =1

4

(c)

€

53 2 = 5 5 EX 3 Simplify completely.

(a)

€

log5 25 ⇒

€

y = log5 25 ← → $ $ $ 25 = 5y Clearly,

€

y = 2


Hint: Do this by switching back and forth between logarithmic form and exponential form.


(b)

€

log2 8 2

(c)

€

log41

64

"

# $ $

%

& ' '

(d)

€

2 log2 7 ⇐ This is exponential form; switch to logarithmic form.

(e)

€

log3 3 (f)

€

logk k


Why?


EX 4 Solve each equation for k. (a)

€

log4 k = 3

(b)

€

log 9 k =3

2

EX 5 Solve each equation for k. (a)

€

logk 81 = 4 (b)

€

logk 5 = −1

Strategy: • Rewrite in exponential form

Strategy:

• Rewrite in exponential form

• Raise both sides to the reciprocal power.


§10 – 5: Laws of Logarithms EX 1 Use the Laws of Logarithms to express in terms of

€

log3 M and

€

log3 N . (a)

€

log3 M 2N 3 ⇐ Use Law 1 ⇐ Use Law 3 twice

(b)

€

log3M

N

"

#

$ $

%

&

' '

3

⇐ Use Law 3 ⇐ Use Law 2

⇐ Rewrite as ⇐ Use Law 3

€

M

€

M 1 2


EX 2 Use the Laws of Logarithms to express each as a single logarithm. (a)

€

4 log M − log N ⇐ Use Law 3 ⇐ Use Law 2

(b) 4 log10 a + 2 log10 b

⇐ Use Law 3, twice ⇐ Use Law 1

(c)

€

3 log 6 m −log 6 n

2

⇐ Rewrite

€

log 6 n2

as

€

12

log 6 n

⇐ Use Law 3, twice ⇐ Use Law 2

If a logarithm does not specify a base, then the base is 10.


EX 3 If

€

log10 4 = 0.60 and

€

log10 3 = 0.48… Find the value by using the Laws of Exponents to rewrite in terms of

€

log10 4 and/or

€

log10 3 (a)

€

log10 36

⇐ Rewrite 36 in terms of 4 and 3:

€

36 = 4 • 32

(b)

€

log101

3000

⇐ How can you rewrite

€

1

3000 in

terms of 4 and 3… and 10…

Yes! You can also use the base!

⇒


EX 4 If log102 = 0.30 and log10 3 = 0.48…

Find the value by using the Laws of Exponents to rewrite in terms of log102

and/or

€

log10 3 (a) log1018

(b) log1020

3

(c) log10 5

(d) log101

23⎛

⎝⎜⎞

⎠⎟


§10 – 6: Applications of Logarithms For this section, you will need a CALCULATOR and… EX 1 Use your calculator to compute the following. (Round your answer to FOUR decimal

places.)

(a) log 4.05 (b) log 0.0065

(c) log3 5.2

EX 2 Use the Definition of a Logarithm and calculator to calculate x to three

significant digits. (a)

€

log x = 3.7135 x = 103.7135 ⇐ Rewrite in exponential form.

(What is the default base?) ⇐ Use your calculator. You may

have a ^,

€

x [] or ke (b) log x = 0.8995 (c) logx = 2.4825

€

10 []

Your calculator will use the “default” base of 10 unless you enter a different base.



EX 3 Solve for x. (a)

€

2 3x = 7

log 23x = log 73x log 2 = log 7

x =log 73 log 2

This is calculator-ready form.

x =

0.84513 0.3010( )

x = 0.9358

(b) 1.5−x = 0.5 (c) 32x = 5

Strategy: If x is in the exponent…

• Rewrite both sides of the equation so they have the same base.

• Then you can use … (This means if the bases are equal, then the exponents are equal.)

In this case, there is no way to rewrite both sides with the same base…

NEW Strategy: If x is in the exponent…

• Take the log of both sides

• Use the Law of Logarithms…

• Divide to get x by itself.


§10 – 7: Problem Solving: Exponential Growth and Decay EX 1 How long will it take an investment of $1,000 to triple in value if it is invested at

an annual rate of 12% compounded quarterly? EX 2 How many dollars must be invested at 16% compounded quarterly to yield

$10,000 at the end of 5 years?

Compound Interest Formula:

P = Principle or amount invested r = annual interest rate in decimal form n = number of times interest is compounded per year t = number of years the investment will grow


EX 3 How much will a $4,000 investment be worth after 5 years if it is invested at 8%

interest compounded quarterly? EX 4 A certain bacteria population doubles in size every 12 hours. By how much will it

grow in 2 days?

Doubling-Time Growth Formula:

= original population size

d = unit of time that the population doubles (can be years, days, hours) t = time (in same units as d)


EX 5 The half-life of carbon-14 (C-14) is 5,730 years. How much of a 10.0mg sample

will remain after 4,500 years? EX 6 The half-life of radioactive radon gas is 3.8 days. How much of 100mg of the gas

will be left after 7 days?

All of these formulas are example of exponential equations that have the form y = bx

Half-Life Decay Formula:

= Original amount

h = half-life t = time


§10 – 8: The Natural Logarithm Function EX 1 Write each equation in exponential form.

(a) ln x = 5 (b) ln 2 = 0.69 EX 2 Write each equation in logarithmic form.

(a) e y = 7

(b) e 23 = 1.95

Natural Logarithm

• is the logarithm with the base, .

• is written as

How did we get ?


Natural Logarithm

Logarithmic Exponential Form Form

e

• Irrational number approximately equal to


The GRAPH of the Natural Logarithm Function

EX 3 Simplify. (a)

€

ln e 2 ⇐ Use Law 3

(b)

€

ln 1

e 2

⇐ Rewrite

€

1

e 2 as

€

e −2

⇐ Use Law 3

Laws of Logarithms

Always…

is not defined


EX 4 Simplify. (a)

€

e ln 5

x = e ln 5 → ln 5 = ln x ⇐ Rewrite using x = e y ↔ y = ln x 5 = x

x = 5 → e ln 5 = 5

(b) e ln 8

⇐ Rewrite using x = e y ↔ y = ln x

(c) e ln e

(d) e ln e2


EX 5 Use the Laws of Logarithms to write each as a single logarithm. (a) ln 5 − ln 4 + ln 12

= ln 54

+ ln 12 ⇐ Use Law 2

= ln 5 • 124

⇐ Use Law 1

= ln 15 (b)

€

2 ln 5 + ln 4 − 3 ⇐ Since ln e = 1 , rewrite 3 as 3 ln e ⇐ Use Law 3, twice ⇐ Use Law 1 ⇐ Use Law 2 EX 6 Solve each equation for x. Leave answers in terms of e. (a)

€

ln x = 2

ln x = 2 ln e

ln x = ln e 2

x = e 2

(b) ln1

x= 2 (c) lnx =

1

3

Strategy:

• Since , rewrite…

⇒ Use Law 3

• If , then


EX 7 Solve each equation for x. Leave answers in terms of natural logs. (a)

€

e 2 x = 9

ln e 2x = ln 92x ln e = ln 9

2x = ln 9

x = 12ln 9 = ln 9

12 = ln 3

x = ln 3

(b) e x +1 = 7

(c) e3x = 27

Strategy:

• Take the ln of both sides

⇒ Use Law 3

⇒ Rewrite

• Get x by itself

CLASS NOTES: §10 – 4 thru §10 – 8mszaleski.weebly.com/.../notes_10_4_thru_10_8.pdf · Algebra...

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