Logarithmic Functions The inverse of the exponential ... - TeacherWeb

Logarithmic Functions The inverse of the exponential function; used for sound, magnitudes of earthquakes among other things and, a long time ago, it made computations of large numbers easier to do. Recall how to find an inverse: 1. Switch the x and y variables 2. Solve for y

Given: y

bx

x

by

something new : y logb x

Graph the following and its inverse: y

4x

X -2 -1 0 1 2

Y 1/16 ¼ 1 4 16

x

4y

y

log 4 x

X 1/16 ¼ 1 4 16

Y -2 -1 0 1 2

Discussion: Exponential function Quadrants 1 and 2 Domain All real numbers Range y>0 x-intercept None y-intercept (0,1)

Logarithmic function 1 and 4 y>0 All real numbers (1,0) none

Graphs of inverse functions are always reflections over the line y=x, and the x and y coordinates are interchanged. Worksheet #1 and #2

Do Now: 1.

3

d

2. m

6

2

d

1 6

4 3

3. Given P(t) 550(1.04)t represents a population. What is theinitial population? What is the growth rate? 4. Jen puts $40 in the bank that earns .8% interest, how much money will she have after 3 years?

A lot of times we are asked to solve logarithmic equations. Most log equations must be written in exponential form before solving. We must learn to rewrite log expressions/equations in exponential form: General rule: logb c

a

ba

c

Use worksheets #3 and #4 for examples and for homework.

Extra examples if needed: Rewrite the given in either log form or exp. Form: 1. 2

log 5 25

2. 3 log 4 64

1 2

log 9 3

4. 1 log10

125

1 6. 3

3.

5. 5 7. 4

3

2

1 16

4

1 10

1 81

8. 10 3 1000

Logarithm Rules: 1. Log of a product: log(xyz) log x log y log z 2. Log of a quotient: log 3. Log of a power:

x y

log x a

log x log y a log x

NOTE: log(xy) is not the same as log(x)log(y) !!!!! And log(x/y) is not the same as log(x)/log(y) !!!!!

Expand using log rules: 1. log

145 12

2. log 5 2

3. log

ab c

4. log

c ab

5. log b 3 6. log 3 x

ex. If logb 3 g and logb 2 h, express the following in terms of g and h only. a. log b 6

b. log b 12

c. log b

3 2

d. log b 36

ex. If logb x

logb p logb t

1 logb q, which 2

exp ression represents x? a. pt q

b. p t q 2

c.

pt q

d.

p tq

ex. If x

m2 n , which expression represents logb x? p s

a. 2 logb m logb n

1 logb s 2

b. 2 logb m logb n logb p

1 logb s 2

1 logb s) 2 d. 2 logb m logb n (logb p logb s) c. 2 logb mn (logb p

ex. Evaluate: log2 32 log2 1

Examples: Express x in terms of a and b only: 1. log 3 a log 3 b

log 3 x

2. log 4 a log 4 b

log 4 x

3. a log 2 b

4. log 6 a

log 2 x

1 log 6 b 2

log 6 x

5. log 3 x log 3 b

log 3 a

6. 2 log a 3log b

log x

7. If log 2 a and log 3 b and log10 1, write the given expressions in terms of a and b. 1. log 6

2. log 9

3. log 20

4. log 30

Common Logs – have a base of 10 and is implied, which means we don’t write log base 10. It’s written as: y log x not y log10 x

Examples: 1. Rewrite log 100,000 = 5 in exponential form.

2. Rewrite 10 2 0.01 in log form.

3. Evaluate the following: a. log1 b. log 0.1 c. log100 d. log 34.6

Natural logs: have a base of the irrational number “e” (2.718…….) Written as y ln x not y loge x Examples: 1. Rewrite ln

1 e

1 in exponential form.

2. Rewrite e0 1 in log form.

3. Evaluate ln 4.6 to the nearest ten-thousandth.

Exponential Equations: Recall with these types of equations, we had to find a common base. Well what would happen if we couldn’t find a common base? Ex. Solve the following: 27 x 9 x we change it to: 7x

1

(looking good so far, but what happens when

9x 1 ?

Discuss what we can’t do: divide by 7, subtract 7, etc…….. Steps for solving exponential equations when no common base can be found: 1. 2. 3. 4.

Take the log of both sides. Using the Power rule for logs, bring the power out front. Solve for x, by dividing through by the log of the number. Make sure you round your answer correctly. (talk about decimal places)

Examples: Solve for x to the nearest hundredth: 1. 7 x

83

log 7 x

log 83

x log 7 log 83 log 83 x log 7 x 2.27

2. 5 x

38

log 5 x

log 38

x log 5 log 38 log 38 x log 5 x 2.26

Solve for x to the nearest tenth: 3. 12 4 x

18

4x

18 12

4x

6

log 4 x

log 6

x log 4 log 6 log 6 x log 4 x 1.3

In the next example, our base is “e”, when this occurs, we ln both sides: Solve for x to the nearest tenth: 4. e.4 x

ln e.4 x ln15 .4x ln e ln15 .4x ln15 ln15 x .4 x 6.8

15

Solve to the nearest ten-thousandth: 5. 1 3e0.52 x

37

3e0.52 x 3e0.52 x e0.52 x e0.52 x

37 1 36 36 3 12

ln e0.52 x ln12 0.52x ln e ln12 0.52x ln12 ln12 x 0.52 x 4.7787

Logarithm Equations – two types: 1. Logs on both sides of the equal sign 2. Logs on one side of the equal sign

Type 1: Use your laws of logs to condense down to one log. Ignore the log part and set the remaining pieces equal and solve for x. Examples: Solve for x and check your answers: 1. log 3 5 log 3 x log 3 (7 x) log 3 5x log 3 (7 x) 5x 7 x 4x 7 7 x 4

2. log(x 3) log x

log 7

log x(x 3) log 7 x2

3x

x2

3x 7

7 0

uh-oh!, quadratic formula!! x

3

9 4 1 ( 7) 2 1

3. log 4 (4x) log 4 x

3

37 2

log 4 64

Now for type 2: Only logs on one side!!

If needed, use your laws of logs to condense more than one log to a single log, making sure to keep the base. Rewrite this log equation in exponential form, this is very important!!! Examples: 1. log 5 7

5x

x

7

x log 5 log 7 log 7 x log 5 x

2. x

4x

log 4 103

103

x log 4 log103 log103 x log 4 x

3. ln x

e2.354 x

2.354

x

These did not need condensing because there was only one log in the equation! However, 4. log x log(x 3) 1 log x(x 3) 1 101

x(x 3)

10

x2

0

x

2

0 x

(x 5)(x 2) 5 and x 2

3x 3x 10

reject the

2

More examples: 1. log 4 x log 4 8 1 x log 4 1 8 x 41 8 32 x

2. log x log 5 2 log 5x 2 10 2 5x 100 5x x 20

3. 4 log 2 3log x 1 log 2 4

solve to the nearest hundredth

log x 3 1

log16x 3 1 101 16x 3 10 x3 16 10 3 3 3 x 16 x

4. 2 log x 3log 5 log x

2

log 5 3 2

x 0 125 x2 10 0 125 2 x 1 125 125 x 2 log

x

0 0

extra problems: 1. log(x 2) log(x 2) 1 2. 2 log 2 x log 2 (x 1)

3

3. log 2 x log 2 (x 4) 5

Change of base formula – used when the base in a log expression is neither 10 nor e. Generally, log a b

Ex. Evaluate: 1. log 6 239

2. log 4 64

3. log 2 .36

log 239 log 6

log 64 log 4

log.36 log 2

log b log a

Some homework problems: Solve for x to the nearest thousandth: 1. x 4

3. log x 7

2

5. log x 36

7. x1.2

2. x 3 107

26

2

11. log x 5

3

6. log ( x

27

8. x 4.3

7

9. log x 107

4. log x 6

3.8

1)

3

26

10. log 4 x log 4 (x 3) 1

2.1

12. If log x = a and log y = b and log z = c, expand: log of a , b and c only.

x y and write it in terms z4

Applications of exponential equations: Recall: f (x) ab x Where a is the initial amount b is the growth or decay rate (1+rate or 1-rate) x is the time f(x) is the amount after time x

Use examples from worksheet #17.

1.

A

Pert

P

3850

e 2.718 r

6%

t

5

A

.06

3850(2.718)(.06 5)

10000

b) find t?

5196.79

5197.00

3850(2.718)(.06t )

10000 2.718.06t 3850 10000 ln .06t ln e 3850 10000 ln .06t 3850 10000 ln 3850 t .06 t 15.9 16years

2. nt

r A P 1 n P 2000 r 6.75% .0675 n 12 t ? A

4000 .0675 2000 1 12

4000

t 11yrs.

3. t

f (t) 80(.5) 60 15% of 80 12

12

80(.5)

12 80

.5 60

t

12 80 12 log 80 log.5

log

t 60

t log.5 60

t 60 12 log 80 t 60 log.5 t 164.2 165 days

(12t )

Review for test: 1. (x 5)2

(y 2)2

9

center

radius

2. graph 3 4i and 2 5i. graph their sum and write in a bi form. 3. Where is y=logx undefined? 4. What is the domain of y=3 x ? 5. What is 5 1 ? 6. What is 5 0 ? 7. What type of roots do the following have: a. x 2

b. x 2

3x 5

7x 3

8. If logx=a and logy=b and logz=c, write log

x2 y3 z

in terms of a,b and c only.

9. Solve for x: lnx=4.3

10. A car costs $13000 new. It has depreciated 30% and is now worth only $3000, how old is the car?

11. Solve for x: log3 (x 5) 4

1

12. Solve for x: (x-1) 2

7

13. Fractional equation problem:

14. Population problem:

15. Solve: log5 x log5 (x 2) 1