Ch. 8 - Logarithmic Functions AnsKey. 8 – Logarithmic Functions Created by Ms. Lee Page 12 of 12 Reference: McGraw-Hill Ryerson 8) The rate at which an organism duplicates is called

Pre-Calculus 12

Ch. 8 – Logarithmic Functions

Created by Ms. Lee Page 1 of 12

Reference: McGraw-Hill Ryerson

First Name: ________________________ Last Name: ________________________ Block: ______


8.1 – UNDERSTANDING LOGARITHMS 2

HW: p. 380 #1, 3 – 8, 12 – 14 4

8.2 – TRANSFORMATIONS OF LOGARITHMIC FUNCTIONS 5

HW: p. 389 #1 – 7, 12 (#12 requires graphing calculator). 5

8.3 – LAWS OF LOGARITHMS 6

HW: p. 400 #1 – 3, 6 – 12 (odd letters, if time, do all letters) 9

8.4 – LOGARITHMIC AND EXPONENTIAL EQUATIONS 10

HW: p. 412 # 1 – 11 12

Pre-Calculus 12




8.1 – Understanding Logarithms

Given an

Exponential

Function,

xxfy 2)( ==

x y

M

M

We want a function that

does just the inverse of

what the exponential

function does,

)(xfy −=

x y

M

M

Given an

Exponential

Function,

xxfy 10)( ==

x y

M

M

We want a function that

does just the inverse of

what the exponential

function does,

)(xfy −=

x y

M

M

Pre-Calculus 12




Definition:

For the exponential function xcy = , the inverse if ycx = . This inverse is also a function and is called

a logarithmic function. It is written as xy clog= , where c is positive number other than 1.

Since our number system is based on powers of 10, logarithms with base 10 are widely used and are

called common logarithms. When you write a common logarithm, you do not need to write the base.

Ex: log x means x10log

Examples:

1. Evaluate each logarithm.

a) 49log7

b) 1000000log

c) 5

9 81log

d) 0001.0log e) 1log2 f) 39log3

2. Determine the value of x .

a) 5log10 =x

b) 416log =x

c) 216log =x

d) 38

1log −=

x

e) 2log4 −=x f)

4

1log16 −=x

g) 3

29log =x

h) 0log3 =x i) 5.0log5 −=x

Pre-Calculus 12




3. Evaluate each expression.

a) x5 , where 10log5=x

b) ))1024(loglog 45

4. Sketch each logarithmic function.

xy 3log=

Equation of the Asymptote:

Domain:

Range:

y-intercept:

x-intercept:

xy2

1log=

Equation of the Asymptote:

Domain:

Range:

y-intercept:

x-intercept:

HW: p. 380 #1, 3 – 8, 12 – 14

Pre-Calculus 12




8.2 – Transformations of Logarithmic Functions Translations of a Logarithmic Function:

khxy c +−= )(log

Reflections, Stretches:

)(log bxay c=

Given the graph of xy 2log= , sketch the graph

of 1)2(log2 −+= xy .


of )3(log2 xy x−= .

Putting together:

khxbay c +−= ))((log

Putting together: Write the equation that describes the red graph,

given xy 4log= in blue.


of 1)62(log2 −+−= xy .

HW: p. 389 #1 – 7, 12 (#12 requires graphing calculator).

Pre-Calculus 12




8.3 – Laws of Logarithms

Laws of Logarithms for Powers

If x and n are real numbers, and 0>x , then

xnx cn

c loglog = where 1,0 ≠> cc

Proof:

Laws of Logarithms for Multiplication

If M and N are positive real numbers, then

NMNM ccc logloglog +=⋅ where 1,0 ≠> cc

Proof:

Pre-Calculus 12




Laws of Logarithms for Quotient

If M and N are positive real numbers, then

NMN

Mccc logloglog −= where 1,0 ≠> cc

Proof:

Examples: Use the laws of logarithms to expand expressions.

1) Write each expression in terms of individual logarithms of ,, yx and z .

a)

z

xy5log

b) xy6log

c) 3 2

39

log

x

d) z

yx5

7log

Pre-Calculus 12




Examples: Use the Laws of Logarithms to Simplify Expressions

2) Write each expression as a single logarithm in simplest form. State the restrictions on the variable.

a)

)log5(log2

1log4 333 xxx +−

Restrictions

b)

)6(log)9(log 22

22 −−−− xxx

Restrictions

Examples: Use the Laws of Logarithms to Evaluate Expressions

3) Use the laws of logarithms to simplify and evaluate each expression.

a) 39log3

b) 2log4log1000log 555 −−

Pre-Calculus 12




c) 2log64log2

16log2 333 +−

d) )27log3

16(log12log2 222 +−

Examples: Solve a Problem Involving a Logarithmic Scale

4) The pH scale is used to measure the acidity or alkalinity of a solution. The pH of a solution is

defined as pH = ]log[ +− H , where ][ +H is the hydrogen ion concentration in moles per litre

(mol/L). A neutral solution, such as pure water, has a pH of 7. Solutions with a pH of less than 7

are acidic and solutions with a pH of greater than 7 are basic or alkaline. The closer the pH is to 0,

the more acidic the solution is. A common ingredient in cola drinks is phosphoric acid, the same

ingredient found in many rust removers. A cola drink has a pH of 2.5. Milk has a pH of 6.6. How

many times as acidic as milk is a cola drink?

HW: p. 400 #1 – 3, 6 – 12 (odd letters, if time, do all letters)

Pre-Calculus 12




8.4 – Logarithmic and Exponential Equations

Examples: Solve Logarithmic Equations

Solve.

1) 12log4loglog 777 =+x

2) )4(log3)6(log 22 −−=− xx

3) 10)8(log 523 =− xx

Examples: Solve Exponential Equations Using Logarithms

Solve. Round your answers to two decimal places.

4) 25002 =x

5) 17005 3=

−x

Pre-Calculus 12




6) 313 86 ++=

xx

Examples: Word problems

7) Paleontologists can estimate the size of a dinosaur from incomplete skeletal remains. For a

carnivorous dinosaur, the relationship between the length, s, in meters, of the skull and the body

mass, m, in kilograms, can be expressed using the logarithmic equation:

4444.3loglog6022.3 −=⋅ ms .

To the nearest hundredth of a meter, what was the skull length of a Tyrannosaurus Rex with an

estimated body mass of 5500 kg?

Pre-Calculus 12




8) The rate at which an organism duplicates is called its doubling period. The general equation is

dt

NtN )2()( 0= , where N is the number present after time t , 0N is the original number, and d is

the doubling period. E. coli is a rod-shaped bacterium commonly found in the intestinal tract of

warm-blooded animals. Some strains of E. coli can cause serious food poisoning in humans.

Suppose a biologist originally estimates the number of E. coli bacteria in a culture to be 1000.

After 90 min, the estimated count is 19 500 bacteria. What is the doubling period of the E. coli

bacteria, to the nearest minute?

HW: p. 412 # 1 – 11

Documents

Ch. 8 - Logarithmic Functions AnsKey. 8 – Logarithmic Functions Created by Ms. Lee Page 12 of 12 Reference: McGraw-Hill Ryerson 8) The rate at which an organism duplicates is called