Lecture 22: Section 3.3 Properties of Logarithms Properties: log (uv









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Lecture 22: Section 3.3 Properties of Logarithms Properties: log (uv

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216921 Lecture 22: Section 3.3 Properties of Logarithms Properties: log (uv

Lecture 22: Section 3.3

Properties of Logarithms

Properties:

log a(uv) = logau+ logav log auv = log aulogav log aun=nlogau

Change of base formula

L22 - 1

Recall the following properties of Logarithm:

The logarithmic function with basea

y=f(x) = logaxif and only if

1. Domain off:

2. log

a1 =

3. log

aa=

4. log

aax= for all real numberx a logax= forx >0

The Natural Logarithmic Function

y= lnxif and only if

Note the following:

ln1 = lne= e lnx= ln(ex) =

L22 - 2

Properties of Logarithms

Letu;vandabe positive real numbers witha6= 1

andnbe any real number. The following properties hold:

1. log

a(uv) =

2. log

auv

3. log

aun=

Proof:

NOTE:loga(u+v)6= logau+ logav

(log au)n6=nlogau

L22 - 3

ex.Evaluate:

1) log

42 + log432

2) log

280log25

3)13 log48 ex.Rewrite and simplify if possible:

1) ln(2 +ex)

2) log

2(xy) 3) log3xlog

3y; y6= 1

L22 - 4

4) ln 13 pe

5) log

9 4p9 3 6) 2

4log2x

7) ln rx 3yz

L22 - 5

ex.Rewrite and simplify:

1) lnpx

3ex1x 2+ 1

2) log

qx py pz

L22 - 6

ex.Write as a single logarithm: 1) 12 [ln(x5) + lnx]ln(2y)

2) log2xlog(x+ 1)13

log(3x+ 7)

L22 - 7

Change of Base Formula

Leta6= 1; b6= 1 andxbe positive real numbers.

Then log ax=logbxlog ba

When using a calculator, we need the specic

formulas: log ax=logxlogaor logax=lnxlna NOTE:

Most calculators have both 'log' and 'ln' keys to

calculate the common and natural logarithm of a number. By using the Change of Base Formula, we can

1. evaluate logarithms to other bases.

2. graph logarithms to other bases.

L22 - 8

ex.Given log50:7;log30:48;ln31:1, and ln(4+e) = 1:9, use Change of Base Formula to nd:

1) log

35

2) log

p3 p4 +e ex.Solve log

3x= log9(2x1)

L22 - 9

Practice.

1) Write log

4x+4log2yas a single logarithm with

base 2.

2) Solve: 2log

3x= log916

3) Solve: log

2x= log425

4) Solve: log

5x= logp5

6Answer.1) log2pxy

42)x= 2 3)x= 5 4)x= 36L22 - 10

Lecture 22: Section 3.3

Properties of Logarithms

Properties:

log a(uv) = logau+ logav log auv = log aulogav log aun=nlogau

Change of base formula

L22 - 1

Recall the following properties of Logarithm:

The logarithmic function with basea

y=f(x) = logaxif and only if

1. Domain off:

2. log

a1 =

3. log

aa=

4. log

aax= for all real numberx a logax= forx >0

The Natural Logarithmic Function

y= lnxif and only if

Note the following:

ln1 = lne= e lnx= ln(ex) =

L22 - 2

Properties of Logarithms

Letu;vandabe positive real numbers witha6= 1

andnbe any real number. The following properties hold:

1. log

a(uv) =

2. log

auv

3. log

aun=

Proof:

NOTE:loga(u+v)6= logau+ logav

(log au)n6=nlogau

L22 - 3

ex.Evaluate:

1) log

42 + log432

2) log

280log25

3)13 log48 ex.Rewrite and simplify if possible:

1) ln(2 +ex)

2) log

2(xy) 3) log3xlog

3y; y6= 1

L22 - 4

4) ln 13 pe

5) log

9 4p9 3 6) 2

4log2x

7) ln rx 3yz

L22 - 5

ex.Rewrite and simplify:

1) lnpx

3ex1x 2+ 1

2) log

qx py pz

L22 - 6

ex.Write as a single logarithm: 1) 12 [ln(x5) + lnx]ln(2y)

2) log2xlog(x+ 1)13

log(3x+ 7)

L22 - 7

Change of Base Formula

Leta6= 1; b6= 1 andxbe positive real numbers.

Then log ax=logbxlog ba

When using a calculator, we need the specic

formulas: log ax=logxlogaor logax=lnxlna NOTE:

Most calculators have both 'log' and 'ln' keys to

calculate the common and natural logarithm of a number. By using the Change of Base Formula, we can

1. evaluate logarithms to other bases.

2. graph logarithms to other bases.

L22 - 8

ex.Given log50:7;log30:48;ln31:1, and ln(4+e) = 1:9, use Change of Base Formula to nd:

1) log

35

2) log

p3 p4 +e ex.Solve log

3x= log9(2x1)

L22 - 9

Practice.

1) Write log

4x+4log2yas a single logarithm with

base 2.

2) Solve: 2log

3x= log916

3) Solve: log

2x= log425

4) Solve: log

5x= logp5

6Answer.1) log2pxy

42)x= 2 3)x= 5 4)x= 36L22 - 10


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