Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions...

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Exponential and Logarithmic Functions 5

Transcript of Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions...

Page 1: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Exponential and Logarithmic Functions

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Page 2: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

5.3Logarithms

Exponential and Logarithmic Functions

Objectives• Switch between exponential and logarithmic form

of equations.• Evaluate logarithmic expressions.• Solve logarithmic equations.• Apply the properties of logarithms to simplify

expressions.

Page 3: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Logarithms

Definition 5.2

If r is any positive real number, then the unique exponent t such that bt = r is called the logarithm of r with base b and is denoted by logb r.

Page 4: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Logarithms

According to Definition 5.2, the logarithm of 16 base 2 is the exponent t such that 2t = 16; thus we can write log2 16 = 4. Likewise, we can write log10 1000 = 3 because 103 = 1000. In general, Definition 5.2 can be remembered in terms of the statement

logb r = t is equivalent to bt = r

Page 5: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Logarithms

Evaluate log10 0.0001.

Example 1

Page 6: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Logarithms

Solution:Let log10 0.0001 = x. Changing to exponential form yields 10x = 0.0001, which can be solved as follows:

10x = 0.0001

10x = 10-4

x = -4

Thus we have log10 0.0001 = -4.

Example 1

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1 10.0001 10

10,000 10

Page 7: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Properties of Logarithms

Property 5.3

For b > 0 and b 1,logb b = 1 and logb 1 = 0

Page 8: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Properties of Logarithms

Property 5.4

For b > 0, b 1, and r > 0,blogb r = r

Page 9: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Properties of Logarithms

Property 5.5

For positive numbers b, r, and s, where b 1,logb rs = logb r + logb s

Page 10: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Properties of Logarithms

If log2 5 = 2.3219 and log2 3 = 1.5850, evaluate log215.

Example 5

Page 11: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Properties of Logarithms

Solution:

Because 15 = 5 · 3, we can apply Property 5.5 as follows:

log2 15 = log2(5 · 3)

= log2 5 + log2 3

= 2.3219 + 1.5850 = 3.9069

Example 5

Page 12: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Properties of Logarithms

Property 5.6

For positive numbers b, r, and s, where b 1,

log log logb b b

rr s

s

Page 13: Exponential and Logarithmic Functions 5. 5.3 Logarithms Exponential and Logarithmic Functions Objectives Switch between exponential and logarithmic form.

Properties of Logarithms

Property 5.7

If r is a positive real number, b is a positive real number other than 1, and p is any real number, then

logb rp = p(logb r)