Logarithms

Rewrite the equations in exponential form.

481log3

17log7

8134 1140

771 32

2

1 5

01log14

532log21

More Logarithms

Simplify the log without a calculator

3

1

125log51

6log36

3

2

1

32log2 3log27

12log 75.0ln

5

08.1

3

1

3.0

64log4

2.log5

x162log10

x162log

Find the log button on your calculator

USING THE CALCULATOR…

Taking Logs to the AxesGraph the function by completing the table. Pick values for x that are easy to evaluate.

x f(x)

xy 2log

0

1

2

34

Try graphing the base 10 function on your calculator: xxf 10log)(

1

2

4

816

xxf 2log)(

xy 2

More Properties of Logarithms

Simplify the expression without a calculator

1000log1010310

1000

16log2242

16

4log10

4

More Properties of Logarithms

Simplify the expression without a calculator

x25log55

2 2log

5? 22

5 43 9log

4? 93

8x255?

x2

42? )3(3 8? 33

x2? 55

x2? 55

More Properties of Logarithms

Because Logarithms come from exponential expressions, they have similar properties.

x5log2 x22 log5log

x

5log2 x22 log5log

x5log2 5log2x

More Properties of Logarithms

Expand:

y

x3

6

5log yx 6

36 log5log

yx 63

66 loglog5log

yx 666 loglog35log

More Properties of Logarithms

Expand:

43log x 4log3log x

xlog43log

More Properties of Logarithms

Condense:

3log2log9log 3

3log)29log( 3

3log2log39log

3

)29(log

3

24log

More Properties of Logarithms

Condense:

12ln3ln34ln

12ln3ln4ln 3

12ln34ln 3

12

34ln

3

9ln

More Properties of Logarithms

€

If logb 3 =1.71 and logb 4 = 2.23, determine...

logb12

logb 9

Solve for x…

€

log3 8 = x

€

3x = 8

€

log3x = log8

€

x log3 = log8

€

x =log8

log3=1.89

Solve for x…

€

log814 = x

€

8x =14

€

log8x = log14

€

x log8 = log14

€

x =log14

log8=1.27

Solving Equations Warm-up

12 279 xx

€

32( )

2x= 33

( )x−1

)1(34 33 xx

)1(34 xx

334 xx

3x

Solving Equations

)5(log)74(log 55 xx

574 xx

123 x

4x

Solving Equations

3)15(log4 x

1543 x

1564 x

x565

13x

Solving Equations

3log)12(log 44 xx

3)12(log4 xx

)12(43 xx

xx 1264 2 64120 2 xx

)16)(4(0 xx16,4 xx

Applications: Sheep populationRetro virus Zappa is decimating the sheep population in the Tungan wastelands. The sheep population is currently 300,000 and is shrinking at a rate of 7% per year

a) Write a function V(t) that will output the number of sheep after t years.

b) Use the function in part (a) to find the sheep population in 10 years.

c) If the trend continues, in how many years from now will the population drop to below 50,000 sheep?

?

ttV 93.300000)(

1093.300000)10( V 7.145194)10( V

t93.30000050000 t93.167.

€

log.167 = log .93( )t

t66.24

Applications: U.S. population?

The population of the United States is currently 310,000,000 people. The population is increasing at a rate of 2% per year.

a) Write a function V(t) that will output the population total after t years.

b) What will be the population of the U.S. in twenty years?

c) If the rate of growth continued, about how long would it take for the population of the U.S. to reach one billion people?

ttV 02.1310000000)(

2002.1310000000)20( V 8.460643692)20( V

t02.13100000001000000000

t02.123.3

€

log3.23 = log 1.02( )t

t21.59

Applications: Car Value Depreciation

A car is purchased for $25,000. It depreciates at the rate of 16% per year.

a) Write a function V(t) that will output the value of the car after t years.

b) Use the function from part (a) to determine how long it will take for the car to be worth $2000.

ttV 84.25000)(

t84.250002000 t84.08.

€

log.08 = log .84( )t

t49.14

Applications: Bank AccountsYou deposit $800 at a bank that pays 4.8% annual interest, compounded monthly.

a) Write a function V(t) that will output the account balance after t-years.

b) Use your function from part (a) to determine what your account balance will be after 12 years. (Round to the nearest $0.01)

c) How long will it take for you to double your money?

ttV 12004.1800)(

)12(12004.1800)12( V 49.1421)12( V

t12004.18001600

t12004.12

€

log2 = log 1.004( )12t

€

173.63 =12t

t47.14