math-1050 session #25 - jefflongnuames.weebly.com€¦ · session #25 6.6: properties of logs (1)...
TRANSCRIPT
Math-1050Session #25
6.6: Properties of Logs(1) Product of Logs(2) Log of a Power(3) Log of a Quotient
Find )(1 xf −
425 += xy
42log 5 += yx
yx 2log4 5 =+−
yx=
+−
2
log4 5
425)( += xxf
425 += yx
xy 5log2
12+−=
Replace f(x) with ‘y’
exchange ‘x’ and ‘y’
Log is the exponent (remember how to
convert between the two?)
Solve for ‘y’
The product of powers → add the exponents
Logarithm: another way of writing the exponent
log2 8 + log2 4
Convert each exponent above into a log:
3 + 2 = 5
= log2 32
23 ∗ 22 = 25
This is the logarithm equivalent of the multiply powers property
of exponents.
23 ∗ 22 = 25
SRRS bbb loglog)(log +=
)5*3(log15log 22 =
Log of a Product Property
log of a product = sum of the logs of the factors.
5log3log15log 222 +=
xy3log yx 33 loglog +
45log 35log3log3log 333 ++
5*3*345 = 5log3log2 33 +
Expand the Logarithm: use properties of logs to rewrite a
single log as an expression of separate logs.
Expand the Logarithm: use properties of logs to rewrite a
single log as an expression of separate logs.
)3log( 2xy 2loglog3log yx ++=
yyx logloglog3log +++=
yx log2log3log ++=
6log 4
xyw2ln
2log3log 44 +=
wyx lnlnln2ln +++=
)5*7(log 2=
xlog5log +
5log7log 22 + 35log 2=
Condense the Logarithm: apply properties of logarithms to rewrite
the log expression as a single log.
x5log=
7log5log 57 + “unlike logs” → can’t condense
“Condense the Log”
7log2log 55 +
4log9log +
4log6log 85 +
14log 5=
36log=
“unlike logs” → can’t condense
9log2
3log3log 22 +=
“Expand the Log”
)3*3(log2=
3log2 2=
Notice something interesting
9log22
2 )3(log= 3log2 2=
Log of a Power Property
16log3
4log4log 33 +=
“Expand the Product”
)4*4(log3=
4log2 3=
Notice something interesting
16log3
2
3 )4(log= 4log2 3=
Log of a Power Property
10log2 5=
2
5 10log
2
5 10log
10log10log 55 +=
10log2 5=
“Expand the Product”
Log of a product is the sum of the logs
of the factors.
Combine “like terms”
New property: “log of a power”
3log x
Use Log of a Power simplify
8ln
xlog3=
32ln= 2ln3=
xlog 21
log x= xlog2
1=
Gotcha’
Which one?y3log5=
53log y5log3log y+=
y3log5 ( )53log y= 553log y=
Log of a Power
c
b Rlogc Rc blog→
A potential error is this:
3
2 6log x x6log3 2=
What is the error ? ‘3’ is an exponent of the base ‘x’ not ‘6x’
Correct the error.
3
22
3
2 log6log6log xx +=
x222 log32log3log ++=
3
2
y
x Using properties of exponents: rewrite this so
the ‘y’ term is NOT in the denominator.32 −yx
2
5log 3
2log)1(5log 33 −+=
( )1
3 2*5log −=
1
33 )2(log)5(log −+=
Negative Exponent Property
Log of a Product Property
Log of a Power Property
2log5log 33 −= Definition of Subtraction: (adding
a negative is subtraction)
Log of a Quotient Property
SRS
Rbbb logloglog −=
2
5log 3
2log5log 33 −“expand the quotient”
3ln8ln − “condense the quotient”3
8ln
“Negative Log”→ denominator of the logarand
16log8log 55 −
5
4log
Condense the quotient
2log5log 44 −
Expand the Quotient
7
3ln
5log4log −=
7ln3ln −=
5log2log2log −+=
5log2log2 −=
2
5log 4=
16
8log 5=
2
1log 5=
Expand the Logarithm
The
denominator
is a product!
53
2log
y
x
)log53(log2log yx +−=
yx log53log2log −−=
Distributive property!
yx log53loglog2log −−+=
53log2log yx −=
Logs of factors in the numerator will be positive.
Logs of factors in the denominator will be negative.
Expand the quotient
3
2
5ln
y
zx 32 5lnln yzx −=
)ln5(lnlnln 32 yxz +−+=
)ln35(lnln2ln yxz +−+=
yxz ln35lnln2ln −−+=2
7
5
4log
=
x
w14
10
4logx
w=
14
4
10
4 loglog xw −=
xw 44 log14log10 −=
yz
x
4
2log4 yzx 4log2log 44 −=
zyx 44444 loglog4loglog2log −−−+=
zyx 44444 loglog4loglog2
12log −−−+=
5log 44log
5log
10
10=
c
aa
b
bc
log
loglog =
Change-of-Base Formula for Logarithms Change to log base 10 or base ‘e’
(your calculator can do these).
Convert to base 10.
6021.0
699.0= 161.1=
4ln
5ln=5log 4
386.1
609.1= 161.1=
=log10 6
log10 3𝑙𝑜𝑔 3 6 =
𝑙𝑜𝑔612
𝑙𝑜𝑔 312
=
12𝑙𝑜𝑔6
12𝑙𝑜𝑔 3
=𝑙𝑜𝑔6
𝑙𝑜𝑔 3= 𝑙𝑜𝑔6 − 𝑙𝑜𝑔3 NO!!!!!
log6
𝑙𝑜𝑔3≠ log
6
3
All non-base 10 logs are vertical stretches of the base 10 log.
xy 2log=
2log
log xy =
change of base formula.
301.0
log x= xlog*32.3=
12log
2log2log 2 ==
x=2log2
1=x
22 =x
Simplify
2log2
Using Change of base:
Simplify: 16log4
2
4 4log 4log2 42)1(2 =
“4 raised to what power equals 16?”
2log2
21
2 2log 2log2
12
“2 raised to what power equals the square root of 2?”
)1(2
12
1
Simplify: 27log5 3
3
3 3log5 3log)5*3( 315
4)16(log6 2 −
4)2(log6 4
2 −
4)2(log)6*4( 2 −
4)1)(24( −
20
Simplify:
3)125(log8 5 +
5)81(log2 9 −
4)3(log6 23 +−
27=
1−=
1=