Aim: What is the natural logarithms?

Do Now:

HW: p.338 # 8,16,20,26,30,38,42,48,50,52,56,58

Given f(x) = ex

write the inverse function

x = ey , we can rewrite x = ey as log form

loge x = loge ey y = loge x

f -1(x) = loge x is the inverse to f(x) = ex

f(x) = logex is called natural logarithms, the notation is f(x) = ln x, or y = ln x

loge x = ln x

Natural logarithms is the log with natural base (base e)

Math Composer 1. 1. 5http: / / www. mathcomposer. com

-5 -4 -3 -2 -1 1 2 3 4 5

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Math Composer 1. 1. 5http: / / www. mathcomposer. com

f(x) = ex

y = x

f(x) = ln x

Properties of natural logarithms

1. ln ex = x 2. eln x = x

3. ln 1 = 0, e0 = 1 4. ln e = 1, e1 = e

5. If ln x = ln y, then x = y

Evaluate the following without calculator

a. e

1ln – 1

b. ln e2 2

c. ln e0 0

d. 2 ln e 2

Simplify:

a. –1 + ln e2x

b. 3ln8 xe

= 2x – 1

= x3 – 8

c. 5 – ln e2x + eln x = 5 – 2x + x = 5 – x

Evaluate the following to the nearest hundredth:

1. 1ln2

13ln

2

1

2.8ln2

ln6ln e

3. 4ln2

14ln2ln2 e

0.19

0.55

-0.08

Find x to the nearest tenth

87534.2ln ee x 87534.2ex

x 17.7

1. ln x = 2.87534

2. ln x = -0.05729

05729.0ln ee x 05729.0exx 0.9

3. ex = 35

35lnln xe x = ln 35 3.6

If ln 2 = x and ln 3 = y. Find the #1 & #2in terms of x and y

1. ln 6

2. 3

1ln

x + y

– y

3. Write as a single

logarithms

3lnlnln2

1zyx

y

zx 3

ln