Unit 4: Exponents and Logarithms (Day 5)

More Laws of Logarithms

Learning

Intention(s): Understand proofs of the Multiplication Law and Power Law

Solve exponential equations

Apply the Change of Base rule to simplify and solve logarithms without base 10

Exponents Logarithms

Multiplication Law: yxyx aaa yxxy aaa logloglog

Division Law: yxyx aaa yx aayx

a loglog)(log

Power Law: xnnx aa )( xnx a

n

a loglog

Proving the Laws:

Multiplication Law

xalog = yalog

then xy =

Power Law

xalog

then nx =

Example: How are log x and log y are related?

i) y = 100x ii) y = 310x

The power law is used to solve exponential equations

1. Isolate the exponential

2. Take the log of both sides

3. Use the power law for logarithms

4. Solve for x

Examples: Solve to 2 decimal places

i) 203 x ii) 480012.1120 n

iii) 352 53 x

Evaluating logs with bases other than 10.

Example: Evaluate to 2 decimal places

i) 37log 4 ii) 15log3

Shortcut: The change of base rule:

Evaluate to 2 decimal places:

80log 2 12log9

12log

37log

4

4 12log

37log

2

2 12log

37log

15

15

So in general:

Use the change of base rule to convert to the same base to simplify

expressions with different bases:

Write as a single log: 9

8

4

4 loglog xx 2

9

3

27 loglog xx

Evaluate: 3log12log 88

What is 3log aa ?

Simplify:

5log3log aaa

5log20log aaa

12log6log2 aaa

Homework:

Worksheet (2.3) # 6, 7, 9, 11, 14-19

Quiz