1) Write in exponential form. log 27 9 = x 3) Evaluate. Warm-Up 2) Write in logarithmic form. 5 x =...
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1) Write in exponential form. log 27 9 = x 3) Evaluate . Warm-Up 2) Write in logarithmic form. 5 x = 2003 log 2 1 8 4) Write the Equation that models this situation: The current world population of Oompa-loompas is 7,986. If the Ooompa-loompa population grows steadily at a rate of 19% per year, what will their total population be
Transcript of 1) Write in exponential form. log 27 9 = x 3) Evaluate. Warm-Up 2) Write in logarithmic form. 5 x =...
1) Write in exponential form.
log279 = x
3) Evaluate.
Warm-Up2) Write in logarithmic form.
5x = 2003
log2
1
84) Write the Equation that models this situation:
The current world population of Oompa-loompas is 7,986. If the Ooompa-loompa population grows steadily at a rate of 19% per year, what will their total population be in 10 years?
8.5 Properties of Logarithms
Properties of Logarithmsloga1 = 0
logaa = 1
logaax = x a xa xlog If loga x= loga y then x = y
because a0 = 1because a1 = a
loglog
loga xx
a 10
10
Change-of-Base
log ( ) log loga a auv u v
log log loga a a
u
vu v
log logan
au n u
Product Property
Quotient Property
Power Property
Remember Common Logarithm??
log10 Is the “common” logarithm
This is the LOG button on the calculator
Sometimes we’re lazy and we don’t even write the base…
log107 = log 7
Basic Properties of Logarithms1) loga1 = 0
2) logaa = 1
because a0 = 1
because a1 = a
a.) log5 1
a.) log3 3
3) One-to-one Property:
If loga x = loga y then x= y
a x.) log log if then ? = ?5 58 2
Product Propertylog ( ) log loga a auv u v
= loga M + loga N1) loga MN
= logb A + logb T2) logb AT
= log M + log A + log T + log H3) log MATH
Express as a sum of logarithms.
Express as a single logarithm
557log = log5 (19*3)
Ex.4) log5 19 + log5 3
5) log C + log A + log B + log I + log N
= log CABIN
Express as a sum of logarithms, then simplify
Ex.
6) log2 (4*16) = log2 4 + log216
= 2 + 4
= 6
Use log53 = 0.683 and log57 = 1.209 to approximate…
Ex. 7
log5 (21)
= log5 3 + log5 7
= 0.683 + 1.209
= 1.892
= log5 (3*7)
Quotient Property
log log loga a a
u
vu v
Express as the difference of logs
log loga aM N
1log
4c 1 4log loga a
Ex.
logaM
N1)
2)
Use log53 = 0.683 and log57 = 1.209 to approximate…
Ex. 3
= log5 3 - log5 7
= 0.683 - 1.209
= -0.526
log5
3
7
Power Property
log logan
au n u
1 334.) log
2 9917.) log
3 7297292.) log
Express as a product.
5log 9b
4 5loga1
4log 5a
= -5 * logb9
Ex.
1
45loga
4)
5)
Use log53 = 0.683 and log57 = 1.209 to approximate…
Ex. 6
= log5 72
= 2(1.209)
= 2.418
log5 49
= 2 log5 7
Ex. 7 log105x3y
log105 + log10y+ log10x3
log105 + log10y+ 3 log10x
Expand.
Expand10
4
23log
•Simplify the division.
104 23log loga
•Simplify the multiplication of 4
10 104 23log log loga
•Change the radical sign to an exponent
1
210 104log log log 23a
•Express the exponent as a product
10 10
1
24 23log log loga
Ex. 8
Ex. Condense.
31
210 10log logx y log log103
10
12
x y
yx310log
2 2log( ) logx x log( ) logx x2 2
log( )x
x
2 2
9)
10)
log log103
10x y
Condense1
5log log log4a a ax y z
•Express all products as exponents
•Simplify the subtraction.
•Change the fractional exponent to a radical sign.
1
45log log loga a ayx z
45log log loga a ay zx
54log loga a
x zy
•Simplify the addition.
45
logazx
y
Ex 11
log log logx y x y
loglog
log
xx y
y
log( ) log logx y x y
Change-of-Base
a
xxa
10
10
log
loglog
this allows us to change from any base to base 10
Ex. 1
Evaluate. Round to four decimal places.
log3 7 log
log10
10
7
3
Be careful about the parenthesis in the calculator…
log(7) / log(3)
= 1.7712
Ex. 2
Evaluate. Round to four decimal places.
log14 2 log
log
2
14= 0.2626
Assignment::
page 496page 496#14-20 even, #14-20 even,
#30-70x4#30-70x4
