logarithms rewrite the equations in exponential form
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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
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