LOGS EQUAL THE

The inverse of an exponential function is a logarithmic function.

Logarithmic Function

x = log a y

read: “x equals log base a of y”

y = bx x = logby

These two equations are equivalent

We can convert exponential equations to logarithmic equations and vice versa, using

this:

3log 5y

2 7loga

logba d

53 y

27a

a db

Convert to exponential form

1)

2)

3)

2 8x2

log 8x1

4y

3

100010

1 log 4y

103 log 1000

Convert to logarithmic form

4)

5)

6)

Now that we can convert between the two forms we can simplify logarithmic expressions. Without a Calculator!

Simplify

1) log2 32

2) log3 27

3) log4 2

4) log3 1

2? = 32

3? = 27

4? = 2

3? = 1

? = 5

? = 3

? = 0.5

? = 0

“What is the exponent of that gives you 32?”

“What is the exponent of 3 that gives you 27?”

Evaluate

6

15) log

36

We can also use these two forms to help us solve for an inverse.The steps for finding an inverse are the

same as before.Easy as 1, 2, 3…1-Rewrite2-Switch x and y3-Solve for y

Example: Find the inverse

1. rewrite (no-need) 8xy 2. Switch and 8yx y x 3. Solve for y 8yx

8xy

Rewrite in log form : 8yx 8 log x y

8 log is the inverse of 8 xy x y

Now you try……..

42xy

2log 4y x

An inverse you just have to know

Ln and are inverses

They undo each other

1.

2.

xe

ln( )xe x

ln( )xe x

Example: Find the inverse

( ) ln( 2)f x x

1. Rewrite: ln( 2)y x

2. Switch and : ln( 2)x y x y

3. Solve for : ln( 2)y x y

ln( 2)

Inverse of ln is

So exponentiate both sides base e: e

x

x y

x e

e

e 2x y

e 2xy

1 e 2xf x

Now you try…..

( ) ln( 4)f x x

Find the inverse:

Essential Question: How do I graph & solve exponential and

logarithmic functions?

Daily Question: How do you expand and condense

logs?

Change-of-Base Formula

loglog

logab

b

xx

a

Let a, b, and x be positive real numbers such that a1 and b1.

loglog

loga xx

a 10

10

logln

lna xx

a

Ex. 1

a) Evaluate using the change-of-base formula. Round to four decimal places.

log3 7 log

log10

10

7

3

Ex. 2You can do the same problem using natural logarithms.

a) Evaluate using the natural logarithms. Round to four decimal places.

log3 7 ln

ln

7

3

Properties of Logarithms

log ( ) log loga a auv u v

ln( ) ln lnuv u v

log log loga a a

u

vu v

ln ln lnu

vu v

ln lnu n un

log logan

au n u

Product Property

Quotient Property

Power Property

Ex. 3log105x3y

log105 + log10y+ log10x3

log105 + log10y+ 3 log10x

Expand.

Ex. 4

ln3 5

7

x

LNMM

OQPPln

( )3 5

7

12x

ln( ) ln3 5 71

2x

1

23 5 7ln( ) lnx

Expand.

Now you Try…..

A. B. C.

Expand each

log32x6y2 5

lnx

yz

32

5 5

2log

xy

z

Ex. 5Condense.

1

23 110 10log log ( )x x log log ( )10

12

1031x x

log log ( )10 1031x x

log [ ( ) ]1031x x

2 2ln( ) lnx x ln( ) lnx x2 2

ln( )x

x

2 2

a)

b)

Now you Try….. Condense each

8 8 8A. log 3 log logx y B. ln 3ln 2ln 4x y

Ex. 6Use and to evaluate the logarithm.

3log 5 1.465 3log 6 1.631

3

6A. log

5

3 3 log 6 log 5

1.631 1.465

0.166

3B. log 30

3 log 5 6

3 3 log 5 log 6

1.465 1.631

3.096

3C. log 36

23 log 6

3 2 log 6

2 1.631

3.262

Ex. 7 Use the properties of logarithms to verify that -ln ½ = ln 2

-ln ½ =

-ln (2-1) =

-(-1) ln (2) =

ln (2) = ln 2

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