Laws of Exponents

Let’s dig deep into our memories and try to remember the laws of exponents.

Product Rule

Quotient Rule nm m nx x x

It’s all coming back now. I remember this stuff.

nm m nx x x

Laws of Logarithms

What are the Laws of Logarithms?Since Logarithms are just exponents

written in a different form, maybe the Laws of Logarithms are the same as the Laws of Exponents.

Product RuleQuotient RulePower Rule

This looks pretty easy, but how does it work with actual numbers?

log ( ) log logb b bx xy y

log ( ) log logb b by x yx

log ( ) lognb bnx x

Examples of the Logarithm Laws

Product Rule

Quotient Rule

Power Rule

May I press the easy button

log ( ) log logb b bx xy y

2 2 2log ( ) l32 32og l8 8og

log ( ) log logb b by x yx

2 2 2log ( ) l32 32og l8 8og

log ( ) lognb bnx x

42 2l 4og (8 8) log

Logarithm Laws in Action

Express in terms of log m and log n3logmn

log x

I can do that.

That one made my brain hurt, but I think I can do it.

12logx g

12

lo x

3logmn

13

logmn

g13

lonm

(lo

13

g log )m n

More Logarithm

Laws in ActionIf log 2 = x and log 3 = y, express each of the following in terms of x and y

6) loga

log 6 32

122log 3

lo12

3g 2

2log2

g1

3lo

)12

( yx

This really isn’t as difficult as I thought it was.

) l 4og2b

log log(24 38 )

32og( 3l ) 3log( ) g2 3lo

2log g33 lo

3 yx

Solving Equations that Contain Logarithms

That sounds confusing, but I guess I’ll give it a try.

If log b A = log b B then A = B

If log N = 2log x + log ysolve for N in terms of x and y

Try to put the equation in the form log b A = log b B

Undo the power law log N = log x2 + log y

Undo the product law log N = log x2y

Drop the log and set the numbers equal

N = x2y

More Equations that Contain Logarithms

Solve for x:

Put the equation in the form log b A = log b B

Undo the power law

Undo the quotient law

Drop the log and set the numbers equal

Evaluate what you can

Solve the equation

log lo 81

g og73

lx

138log l g og7o lx

log log og2 7lx

72

log logx

72x

14x

Exponentiating Logarithmic Equations

Solve for x:Logarithms are exponents written in a different

form Undo the product law

Evaluate what you have

Write in exponential form

I’m not even sure what that means.

You must check for extraneous roots

Write your final answer x = 5

That looked hard at first, but it really wasn’t hard after all.

Let’s give it a look.

4 4log ( ) log ( 2)3 3x x

4 3 3 2log ( )( )x x 2( )3 ( )3 4x x

( )3 6( )3 1x x 2 9 16x

2 25x 5x