MAC1105Logarithmic Functions Final Draft Sec PDF

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MAC1105Logarithmic Functions Final Draft Sec...

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Logarithmic Functions Definition: Let a and x be positive numbers, denoted by and is defined as follows:

MAC 1105

. The logarithm of x (argument) with base a is

if and only if

.

Therefore, the logarithm is an exponent. The expression is read as “logarithm (or log) base a of y equals x.” Especial bases in logarithms Common logarithmic function (means base = 10):

instead of

write

Natural logarithmic function (means base = e):

instead of

write

EXAMPLE Write the logarithmic equation in its equivalent exponential form.

YOUR TURN Write the logarithmic equation in its equivalent exponential form.

1a. log m p = n

1b. log r t = s

2b. log3

2a. log 2 64 = 6

EXAMPLE Write the exponential equation in its equivalent logarithmic form.

1 = −3 27

YOUR TURN Write the exponential equation in its equivalent logarithmic form.

3a. 8 3 = 4

3b. 52 = 25

4a. r s = t

4b. m n = p

2

MDC Mathematics 1

EXAMPLE Evaluate the expressions without using a calculator.

YOUR TURN Evaluate the expressions without using a calculator.

5a. log 2 16

5b. log 3 81

6b. ln e2

6a. log1000

7a. log 5

8a. ln

1 125

7b. log0.1

1

1 10

8b. log 7 1

3 2 e

EXAMPLE Use technology to approximate the value of the following expressions. Round your answer to 4 decimal places.

YOUR TURN Use technology to approximate the value of the following expressions. Round your answer to 4 decimal places.

9a. log123

9b. ln 2

10a. log 17.4 17.4

10b. log32 2

Properties of Logarithms 1. loga 1 = 0, because a 0 = 1

3. loga a x = x , because a loga x = x

2. loga a = 1, because a1 = a

4. If log a x = log a y, then x = y MDC Mathematics 2

EXAMPLE Evaluate the expressions using properties of logarithms.

YOUR TURN Evaluate the expressions using properties of logarithms.

11a. 5 log5 9

11b. log4 411

12a. ln1

12b. log17.4 17.4

Graphing Logarithmic Functions Sketch the graph of f (x ) = 2 x and g (x ) = log 2 x . Determine the x intercept, y intercept, asymptote, domain and range for both functions.

x -3 -2

-1 0 1

f (x ) = 2 x

f (x ) = 2 x y (x, y ) 1 ! 1" # -3, $ 8 % 8& 1 ! 1" # -2, $ 4 % 4& 1 ! 1" # -1, $ 2 % 2& 1 ( 0,1) 2 (1,2 )

2

4

3

8

(2,4 ) ( 3,8)

g (x ) = log2 x x

1 8 1 4

y -3 -2

1 2

-1

1

0

2

1

4

2

8

3

( x, y ) !1 " $ , −3 % &8 ' !1 " $ , −2 % &4 ' 1 ! − " $ , 1% &2 '

(1,0 ) ( 2,1) ( 4,2 ) (8,3 )

g (x ) = log 2 x

x intercept: None y intercept: (0,1 )

x intercept: (1,0) y intercept: None

asymptote: y = 0 (horizontal) Domain: ( −∞, ∞ )

asymptote: x = 0 (vertical)

Range: (− ∞, ∞)

Range: (−∞, ∞ )

Domain: (0, ∞)

MDC Mathematics 3

EXAMPLE Sketch the graph of each function. Determine the x-intercept, y-intercept, asymptote, domain and range for the function. 13a. f ( x ) = log3 (1 − x )

!

! = ! ! = log ! (1 − !)

YOUR TURN Sketch the graph of each function. Determine the x-intercept, y-intercept, asymptote, domain and range for the function. 13b. f ( x) = log( 2 x) + 1

!, ! !

! = ! ! = log 2! + 1

!, !

0 0 1 1 2 2 -1 -1 -2 -2

MDC Mathematics 4

Finding the Domain of Logarithmic Functions: Given the function

, its domain will be given by the values that

satisfy the inequality.

EXAMPLE Find the domain of the logarithmic function.

YOUR TURN Find the domain of the logarithmic function.

14a. y = log( 2 x − 7)

x " 14b. y = log 5 #! $ & x − 3'

MDC Mathematics 5

Additional Exercises Evaluate each expression. You might need technology to approximate the value in some cases. 1.

log125 5

4.

log

7.

2.

log 0.001

3.

ln e

1 243

5.

3 log 1 e−

6.

log 1

log0.01 100

8.

log8

9.

ln π

3

10. log10− n

e

11. e ln1.7

1 144 12

12. log23 1

Sketch the graph of each function. Find the domain and range. 13. f ( x ) = l n (x + 3 ) − 2

14. g ( x ) = log 2 (− x ) + 3

Find the domain of each function. 15. h ( x ) = log (3x − 7 ) − 5

16. m (x ) = log 4 (12 − 5x ) +

1 2

MDC Mathematics 6...