Y =logb x - FRC Precaculus 40S

8.1 Understanding Logarithms

R7

(p. 370-379)

The logarithmic function is the inverse of the exponential function.

Remember, to find the inverse of a function we switch the x and y values and

then solve for y.

Exponential function

y= bx

Inverse function

x = bY

Notice that the y-value is now an exponent.

In order to isolate and manipulate exponents, we must use something called the logarithm function.

y =logb x

where

b base of the log

y the logarithm (the answer) x ----> the argument

Exl: Sketch the graph of y = 2x and its inverse. 1

--

-1

1 2')< X Cf

Z

I 2.. 0

Q.1-k X

If

2.

S-14itck X .41

Note: The equation of the graph of the inverse is y = log2 X

Page 2

Ex2: Express m = 4" in logarithmic form.

109 Li hfl h

Itlypikhvc

Ex3: Express log2 8 = 3 in exponential form.

WE ktow fids

Lx_

fric_tv

Ex4: Evaluate he following expressions:

108 a) log2 16 = x

x

a N.NP61::

b) log2

dcl lr

hol -them

2"

X

0 log3 (-27)

het+ posJal (es to ce_i CU

t \r-AVA4 ?

h 0 soh-)

Thus, logB A = C where A > 0, B> 0, and B I

Note: The base of a logarithms cannot be negative. The argument (A) of a logarithm is always positive.

Page 3

Ex5: Solve the following equations: 1

a) logx 5 =

b) log x = --3 I 0

0 0

e=niti

)e.

Note: When the base is not indicated, this means that there is a base of 10. logx = log10 x

Some Basic Logs to Remember: "Quicksnappers"

a) log 1 = 0

c_x

---

b) logc=

-

c) logc cY =

`0

X

x

2,1

c,

e) C y --

"7:-- CA,_

0 IiN

A--tP

c174

Page 3

Try these; Evaluate

iii) log10 \

1(1

Ex6: Estimate the following value:

1 g 30

C6 n

--a-

OLI

Homework: Page 380 #1-5, 7-10, 13-15

Page 4

8.2 Transformations of Logarithmic Functions

R9

(p. 383-391)

Exl : Sketch the graphs of the functions y = 3x and y = log3 x .

Note that these two functions are inverses of each other.

vo(A. f Eiv k).

10(1 Ab ,

2

Note: The graph of Y = log3 x has a vertical asymptote at x = 0 because x > 0 is a restriction of the argument.

Page 6

Ex2: Sketch the graphs of the following functions on the same Cartesian plane.

a) y = log, x

b) y = log4 x

y = log x d) Y =

Note: All the graphs pass through the point (1,0). The base of the logarithm determines the next point.

Base 2-->(2,l) Base 4 (4,1) Base 10 --> (10,1)

y = log2 x 2Y = x

2 _> (2,1)

Page 6

\90V 5171p

Ex3: Sketch the graph of the function Y = 1,0k5 + 2)-1. (PR 2) down ?

State the domain: Ex>

or E21 ot) )

N

109 5 6A + 2) -

Determine the y-intercept:

y 109 5 (pi 2.) 1

0 B1318 \60/

y lv;

( y a 5.7

)(+2>() -Z ?

ato

t (1 o f

bin( l'Atortrel

Page 8

xt3) ((x-3)

Ex4:

Sketch

the

graph

of

the

function

.3)

=

10 base

3

-- x) + op /

1o? rvtehrerdo- yi\1s.,1.1.31_

3

-2

x c )

3- 1 ') 3 3 -' -3

3 =

x -22'3

Equation of the asymptote:

>< =3

Homework: Page 389 #1, 3-9, 15

IQ33 (3-6) + 1 y 193 3 +- I

Page 9

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