HW Check HW 6.5 1. a. 12,000 bacteria

b. 96,000 bacteriac. 768,000 bacteria

2. 0.5 g3. a. 6,800 insects

b. 425 insects4. 15 hours5. 72 billion6. a. 37,402,600 bact.

b. 6,553,790 bact.

7. 2,609,610 people8. a. 28.6 million

b. 29.1 million9. a. 689 bacteria

b. 2986 bacteria10.a. 172,800 termites

b. 371,806 termitesc. 627 termites

NATURAL LOGARITHMS

6.6a

The Constant: ee is a constant very similar to π.Π = 3.141592654…e = 2.718281828…Because it is a fixed number we can find e2

e3

e4

Exponential Functions with e

Exponential Functions with a base of e are used to describe CONTINUOUS growth or decay.

Natural LogarithmicWe call a log with a base of 10 “Common Log”We can call a log with a base of e “Natural Log”

Natural Log is denoted with the “LN”All the same rules and properties apply to natural log as they do to regular logs

Natural Logs

Natural logs have a base of e.

loge5 Ln5

But instead of writing loge5, we change it to ln5

Natural Logs

A natural logarithm is defined as follows:

If y = ex, then Lny = x

Exponential to Log form1. ex = 6

2. ex = 25

3. ex + 5 = 32

Log to Exponential Form 1. Ln 1 = 0

2. Ln 9 = 2.197

3. Ln (5.28) = 1.6639

Solving Exponential Equations1. ex = 18

2. ex+1 = 30

3. e2x = 12

Solving Logarithmic Equations 4. Ln x = -2

5. Ln (2m + 3) = 8

6. 1.1 + Ln x2 = 6

PROPERTIES OF LOGARITHMS

6.6b

You’ve gotten good at solving exponential equations with logs…

… but how would you handle something like this?

733 4log2log x

Properties of Logarithms

1. Product Property2. Quotient Property3. Power Property

The log bases must be the same in order to apply these properties!

Product Property

NMMN bbb logloglog

Quotient Property

NMNM

bbb logloglog

Power Property

MxM bx

b loglog

Steps for Condensing

Step 1: Put everything in the sky (create exponents).

Step 2: Combine.

Example 1: Condense the following logarithmic expression.

4log20log 33

Example 2: Condense the following logarithmic expression.

yx 22 loglog3

Example 3: Condense the following logarithmic expressions.

16log4log2log3

9log9log3 62

Simplify4. 3 Ln 5

5. Ln 5 + Ln 4

6. Ln 20 – Ln 10

7. 4 Ln x + Ln y – 2 Ln z

Steps for Expanding

Step 1: Separate the combinations.

Step 2: Get everything out of the sky. (move the exponents)

Example 4: Expand the following logarithmic expression.

yx

5log

Example 5: Expand the following logarithmic expression.

43log r

Example 6: Expand the following logarithmic expressions.

2

3log

y

437log ba

Expand7. Ln (xy2)

8. Ln(x/4)

9. Ln(y/2x)

Quick Recap!

To combine:

1. Put everything in the sky.

2. Combine.

To expand:

1. Separate the combinations.

2. Get everything out the sky.