properties of logarithms
DESCRIPTION
Properties of Logarithms. They’re in Section 3.4a. Proof of a Prop ‘o Logs. Let. and. In exponential form:. Let’s start with the product of R and S :. A Prop ‘o Logs!!!. Properties of Logarithms. Let b , R , and S be positive real numbers with b = 1, and c any real number. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/1.jpg)
Properties of Properties of LogarithmsLogarithmsThey’re in Section 3.4aThey’re in Section 3.4a
![Page 2: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/2.jpg)
Proof of a Prop ‘o Logs
Let logbx R and logby SIn exponential form:
yb Sxb RLet’s start with the product of R and S:
x yRS b b x yRS b
logb RS x y log logb bR S A Prop ‘o Logs!!!A Prop ‘o Logs!!!
![Page 3: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/3.jpg)
Properties of Logarithms
Let b, R, and S be positive real numbers with b = 1, and cany real number.
log log logb b bRS R S • Product Rule:
log log logb b b
RR S
S • Quotient Rule:
log logcb bR c R• Power Rule:
![Page 4: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/4.jpg)
Guided PracticeAssuming x and y are positive, use properties of logarithmsto write the given expression as a sum of logarithms ormultiples of logarithms.
4log 8xy 4log8 log logx y 3 4log 2 log logx y
3log 2 log 4logx y
![Page 5: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/5.jpg)
Guided PracticeAssuming x is positive, use properties of logarithms to writethe given expression as a sum or difference of logarithmsor multiples of logarithms.
2 5ln
x
x
1 22 5lnx
x
1 22ln 5 lnx x
21ln 5 ln2
x x
![Page 6: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/6.jpg)
Guided PracticeAssuming x and y are positive, use properties of logarithmsto write the given expression as a single logarithm.
5ln 2lnx xy 25ln lnx xy
5 2 2ln lnx x y 5
2 2lnx
x y
3
2lnx
y
![Page 7: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/7.jpg)
Of the eight relationships suggested here, four are true and fourare false (using values of x within the domains of both sides ofthe equations). Thinking about the properties of logarithms,make a prediction about the truth of each statement. Then testeach with some specific numerical values for x. Finally, comparethe graphs of the two sides of the equation.
1. ln 2 ln ln 2x x
3. 2 2 2log 5 log 5 logx x
5.log
log4 log 4
x x
7. 25 5 5log log logx x x
2. 3 3log 7 7 logx x
4. ln ln ln 55
xx
6.3
4 4log 3logx x
8. log 4 log 4 logx x
These four statements are TRUE!!!These four statements are TRUE!!!
![Page 8: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/8.jpg)
A few more problems…Assuming x and y are positive, use properties of logarithms towrite the expression as a sum or difference of logarithms ormultiples of logarithms.
4log1000x 4log1000 log x 3 4log x
3
25ln
x
y
1 3 2 5ln lnx y 1 2ln ln3 5
x y
![Page 9: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/9.jpg)
And still a few more problems…Assuming x, y, and z are positive, use properties of logarithms towrite the expression as a single logarithm.
14log log
2y z 4log logy z
4
logy
z
3 2 23ln 2lnx yz yz
9 3 6 2 4ln lnx y z y z 9 5 10ln x y z
![Page 10: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/10.jpg)
Whiteboard…Whiteboard…Write as a single logarithmic expression:
2log 100 log3 5logx x22log3 5logx x
52 2log 3 logx x 2 10log9 logx x 2
10
9log
x
x
8
9log
x
![Page 11: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/11.jpg)
Let’s do an exploration…How do we evaluate ?
4log 7
First, switch to exponential form.4 7y Apply ln to both sides.ln 4 ln 7y Use the power rule.ln 4 ln 7y
Set equal to y:4log 7y
Divide by ln4.ln 7
1.404ln 4
y
We just provedWe just provedthe C.O.B.!!!the C.O.B.!!!
![Page 12: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/12.jpg)
Change-of-Base Formula for Logarithms
For positive real numbers a, b, and x with a = 1and b = 1, log
loglog
ab
a
xx
b
Because of our calculators, the two most common forms:
log lnlog
log lnb
x xx
b b
![Page 13: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/13.jpg)
Guided PracticeEvaluate each of the following.
1. 3log 16ln16
ln3 2.524
2. 6log 10log10
log6
1
log6 1.285
3. 1 2log 2 ln 2
ln 1 2 ln 2
ln 2
1
![Page 14: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/14.jpg)
Guided PracticeWrite the given expression using only natural logarithms.
1. 5log 3xln 3
ln5
x
2. 7log 2x y ln 2
ln 7
x y
![Page 15: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/15.jpg)
Guided PracticeWrite the given expression using only common logarithms.
1. 4log slog
log 4
s
2. 1 4log 2a b
log 2
log 1 4
a b
log 2
log 4
a b
![Page 16: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/16.jpg)
Graphs of Logarithmic Functions with Base b
Rewrite the given function using the change-of-base formula.
logbg x x ln
ln
x
b
1ln
lnx
b
Every logarithmic function is a constant multiple of theEvery logarithmic function is a constant multiple of the natural logarithmic function!!!natural logarithmic function!!!
If b > 1, the graph of g(x) is a vertical stretch or shrink of thegraph of the natural log function by a factor of 1/(ln b).
If 0 < b < 1, a reflection across the x-axis is required as well.
![Page 17: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/17.jpg)
More Guided PracticeDescribe how to transform the graph of the natural logarithmfunction into the graph of the given function. Sketch the graphby hand and support your answer with a grapher.
1. 5logg x x 1ln
ln 5x
10.621
ln5 Vertical shrink by a factorVertical shrink by a factor
of approximately 0.621.of approximately 0.621.
How does the graph look???How does the graph look???
![Page 18: Properties of Logarithms](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813d58550346895da71f4b/html5/thumbnails/18.jpg)
More Guided PracticeDescribe how to transform the graph of the natural logarithmfunction into the graph of the given function. Sketch the graphby hand and support your answer with a grapher.
2. 1 4logh x xln
ln1 4
x
10.721
ln 4 Reflect across Reflect across xx-axis, Vertical-axis, Vertical
shrink by a factor of 0.721shrink by a factor of 0.721
How does the graph look???How does the graph look???
1ln
ln 4x