Properties of Properties of LogarithmsLogarithmsThey’re in Section 3.4aThey’re in Section 3.4a

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!!!

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:

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

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

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

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!!!

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

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

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

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.!!!

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

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

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

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

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.

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???

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