y ( y log x x a ) a is equivalent to f(x) = log a x - - PowerPoint PPT Presentation

Lesson 5.2: Logarithmic Functions and Their Graphs

If y = ax, then the inverse is x = ay An equivalent equation for x = ay is y = logax

f(x) = logax Logarithmic Function

( log ) y x x a

a y

  

“is equivalent to”

Ex 1: Evaluate. a)f(x) = log2 x, when x = 32 b)f(x) = log3 x, when x = 1 c)f(x) = log4 x, when x = 2

y x x a

a y

   log

32 2 

y

y  5

y  log2 32 y  log31

1 3 

y

y  0

y  log4 2

2 4 

y

y  1 2

Common Log Function: f(x) = log10 x Ex 2: Evaluate f(x) = log10 x when a) x = 10 c) x = 1/3 b) x = 2.5 d) x = -2

f x ( ) log 

1010

f x ( ) 1 f x ( ) log . 

10 2 5

f x ( ) .  0 398

f x ( ) log 

10

1 3

f x ( ) .477  0 f x ( ) log ( )  

10

2 No al solution Re

Properties of Logarithms

1 1 2 1 3 4 .) log .) log .) log .) log log , .

log a a a x x a a

a a a x If x y then x y

a

     

( ) Because a 1  ( ) Because a a 

1

Ex 3: Use the properties of logs. a)Solve for x. log2 x = log23 b)Solve for x. log4 4 = x c)Simplify. log55x

d)

log

6

6 20

If x y then x y

a a

log log , .  

x  3

loga a 1

x 1

loga

x

a x 

a x

a x

log



x

20

Natural Log Function

a e b e ) ln )

ln

1

5

When f(x) = ex, then the inverse is x = ey.

f(x) = loge x = ln x

Ex 4: Use log properties to simplify.

c d e ) ln ) ln 1 3 2





loge e 1

 1  e

e

log 5

 5

 0 3  0

 2 1 ( )  2

Ex 5: Find the value of x for f(x) = ln x. a)x = 2 b) x = 0.3 c) x = -1 d) x = 1 + sqrt(2) Homework: p.386 #1-29 odd

 ln2  0 693 .  ln . 0 3  1204 .   ln( ) 1

No al solution Re

  ln( ) 1 2  0 881 .