Warm-Up 4/30 The Park family is saving for their son's education. If they deposit

$31,500 in an account earning 7.6%, compounded continuously, how

much will be in the account when Sam goes to college in 9 years?

What if it is compounded annually?

Answer: $62,426.36 $60,900.52

11.4 Logarithmic Functions

The inverse of y = bx is _______The function x = by is called a___________It is usually written ______________Read “ y equals log base b of x” Logarithmic functions are the inverse of exponential functions

x = by

logarithm

y = logbx

Definition:y= logbx if and only if x=by

“b” can’t be 1 and it must be positive

EX1: Write in exponential form

a) log273 = 1/3 b) log164 = ½

Answers:

EX2: Write each equation in logarithmic form

a) 210 = 1024 b) 2-3 = 1/8

Answers: log2 1024 = 10 b) log2 1/8 = -3

1 1

3 227 3 16 4

Ex3 Evaluate: log51/625

This is a number, its an operation The answer to a log will be an exponentThink 5 to the what power is 1/625Since it is a fraction the exponent will be negative5 4 = 625 so 5 –4 =1/625So log51/625 = -4

Ex 4: evaluate log432Think 4 to the what equals 32 Nothing – dang itRe-write: 4x = 32Get the bases the same: (22)x = 25

Bases are same so just set exponents equal to each other2x = 5X = 2.5

Since a log is inverse of an exponent it follows the exponent rules…

m and n are positive numbers, b is a positive number other than 1 and p is any real number…

Property DefinitionProduct logbmn =logbm +logbn

Quotient logbm/n = logbm – logbn

Power logbmp = p(logbm)

Power of equality

If logbm=logbn then m=n

Ex 6 Solve:log10 (2x+5) = log10(5x-4)

Which property can I use? Power of equality… the bases are the same and they are equal so2x+5 = 5x – 4 easy 9 = 3x x = 3 are they all this easy – of course not you silly geese.

Ex 7: Solve log3(4x+5) – log3(3 – 2x) =

2 Don’t have logs on both sides so we can’t use the equality property. Always try to simplify – subtraction, write it as a quotient

Re-write using definition of logsnow solve /cross multiply

27 – 18x = 4x + 5-22x=-22x=1

2 4 53

3 2

x

x

3

4 5log 2

3 2

x

x

9(3 2 ) 4 5x x

Ex8: log3(x+2)+log3(x-6) = 2

Write as a single log:Use log properties:

No logs on both sidesWrite in exponential form

Solve: This is a QuadraticYou should know how to solve

CHECK in original equationYou might need to eliminate an answerCan’t take the log of a neg #

log3(x+2)(x – 6)=2

32 = (x+2)(x – 6)

9 = x2 – 4x – 12 0 = x2 – 4x – 21

(x – 7)(x + 3)=0

x = 7 x = -3

Ex 9: ½ log8(x+1) – ½ log825 = log84

Use your properties to write as a single log on each side

14

5

x

Subtraction means division

8 81/2

1log log 4

25

x

1 400

399

x

x

Cross multiply and solve

1 20x Square both sides

Summary:Homework: pg 723 # 20-52