3.3 PROPERTIES OF LOGARITHMS

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• Use the change-of-base formula to rewrite and

evaluate logarithmic expressions.

• Use properties of logarithms to evaluate or

rewrite logarithmic expressions.

• Use properties of logarithms to expand or

condense logarithmic expressions.

• Use logarithmic functions to model and solve

real-life problems.

What You Should Learn

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Change of Base

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Change of Base

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Example 1 – Changing Bases Using Common Logarithms

a.

b.

Use a calculator.

Simplify.

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Properties of Logarithms

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Properties of Logarithms

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Example 3 – Using Properties of Logarithms

Write each logarithm in terms of ln 2 and ln 3.

a. ln 6 b. ln

Solution:

a. ln 6 = ln (2 3)

= ln 2 + ln 3

b. ln = ln 2 – ln 27

= ln 2 – ln 33

= ln 2 – 3 ln 3

Rewrite 6 as 2 3.

Product Property

Power Property

Rewrite 27 as 33.

Quotient Property

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Example

Find the exact value of each expression without using a

calculator.

a. log _5 53

= log5 513 =

1

3log5 5 =

1

3

b. ln 𝑒6 − ln 𝑒2

= 6 ln 𝑒 − 2 ln 𝑒 = 6 − 2 = 4

𝑜𝑟 = ln𝑒6

𝑒2= ln 𝑒6−2 = ln 𝑒4 = 4 ln 𝑒 = 4

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Rewriting Logarithmic Expressions

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Example 5 – Expanding Logarithmic Expressions

Expand each logarithmic expression.

a. log4 5x3y

b.

Solution:

a. log4 5x3y = log4 5 + log4 x3 + log4 y

= log4 5 + 3 log4 x + log4 y

Product Property

Power Property

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Example 5 – Solution

b.

Rewrite using rational

exponent.

Quotient Property

Power Property

cont’d

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Example

Condense each logarithmic expression.

a.1

2log 𝑥 + 3 log(𝑥 + 1)

= log 𝑥12 + log 𝑥 + 1 3

= log 𝑥12 𝑥 + 1 3

= log 𝑥 𝑥 + 1 3

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Continue…

b. 2 ln(𝑥 + 2) − ln 𝑥

= ln 𝑥 + 2 2 − ln 𝑥

= ln𝑥 + 2 2

𝑥

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Continue…

c. 1

3[log2x + log2 x + 1 ]

=1

3log2 𝑥(𝑥 + 1)

= log2 𝑥 𝑥 + 113

= log2 𝑥(𝑥 + 1)3

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Application

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Application

If the points are graphed and fall on a line, then you can

determine that the x- and y-values are related by the

equation

ln y = m ln x

where m is the slope of the line.

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Example 7 – Finding a Mathematical Model

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Example 7 – Solution

Planets Near the Sun

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Example 7 – Solution

𝑚 =0.632 − 0

0.421≈ 1.5 =

3

2

cont’d

The equation of the line:

Y=3

2X, where Y = ln x and X = ln x

ln 𝑦 =3

2ln 𝑥