Beginning with an In-Class Activity

Section 6-5Properties of Logarithms

Logarithm of 1

Logarithm of 1

For any base b,

logb 1 = 0

Logarithm of 1

For any base b,

logb 1 = 0

Why?

Logarithm of 1

For any base b,

logb 1 = 0

Why?

logb 1 = 0

Logarithm of 1

For any base b,

logb 1 = 0

Why?

logb 1 = 0

b0

Logarithm of 1

For any base b,

logb 1 = 0

Why?

logb 1 = 0

b0 = 1

Logarithm of a Product

Logarithm of a ProductFor any base b and for any positive real numbers x and y,

logb(xy) = logb x + logb y

Logarithm of a ProductFor any base b and for any positive real numbers x and y,

logb(xy) = logb x + logb yWhy?

Logarithm of a ProductFor any base b and for any positive real numbers x and y,

logb(xy) = logb x + logb yWhy?

logb x = m logb y = n

Logarithm of a ProductFor any base b and for any positive real numbers x and y,

logb(xy) = logb x + logb yWhy?

logb x = m logb y = n

bm = x b

n = y

Logarithm of a ProductFor any base b and for any positive real numbers x and y,

logb(xy) = logb x + logb yWhy?

logb x = m logb y = n

bm = x b

n = y

xy = bmibn

Logarithm of a ProductFor any base b and for any positive real numbers x and y,

logb(xy) = logb x + logb yWhy?

logb x = m logb y = n

bm = x b

n = y

xy = bmibn

= bm+n

Logarithm of a ProductFor any base b and for any positive real numbers x and y,

logb(xy) = logb x + logb yWhy?

logb x = m logb y = n

bm = x b

n = y

xy = bmibn

= bm+n

logb xy = m+n

Logarithm of a ProductFor any base b and for any positive real numbers x and y,

logb(xy) = logb x + logb yWhy?

logb x = m logb y = n

bm = x b

n = y

xy = bmibn

= bm+n

logb xy = m+n

logb x = m logb y = n

Logarithm of a ProductFor any base b and for any positive real numbers x and y,

logb(xy) = logb x + logb yWhy?

logb x = m logb y = n

bm = x b

n = y

xy = bmibn

= bm+n

logb xy = m+n

logb x = m logb y = nSo,

Logarithm of a ProductFor any base b and for any positive real numbers x and y,

logb(xy) = logb x + logb yWhy?

logb x = m logb y = n

bm = x b

n = y

xy = bmibn

= bm+n

logb xy = m+n

logb x = m logb y = n

logb(xy) = logb x + logb ySo,

Logarithm of a Quotient

Logarithm of a Quotient

For any base b and for any positive real numbers x and y,

logb

xy( ) = logb x − logb y

Logarithm of a Quotient

For any base b and for any positive real numbers x and y,

logb

xy( ) = logb x − logb y

Why?

Logarithm of a Quotient

For any base b and for any positive real numbers x and y,

logb

xy( ) = logb x − logb y

Why?

Well, now, that’s a homework problem!

Logarithm of a Power

Logarithm of a Power

For any base b, and real positive real number x, and any real number p,

logb x p = p logb x

Logarithm of a Power

For any base b, and real positive real number x, and any real number p,

logb x p = p logb x

Why?

Logarithm of a Power

For any base b, and real positive real number x, and any real number p,

logb x p = p logb x

Why?

It’s another homework problem!

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

= log6(2i3)

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

= log6(2i3)

= log6(6)

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

= log6(2i3)

= log6(6)

6x = 6

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

= log6(2i3)

= log6(6)

6x = 6

x = 1

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

= log6(2i3)

= log6(6)

6x = 6

x = 1

log6 2+ log6 3 = 1

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

= log6(2i3)

= log6(6)

6x = 6

x = 1

= log5

2008( )

log6 2+ log6 3 = 1

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

= log6(2i3)

= log6(6)

6x = 6

x = 1

= log5

2008( )

= log5(25)

log6 2+ log6 3 = 1

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

= log6(2i3)

= log6(6)

6x = 6

x = 1

= log5

2008( )

= log5(25)

5x = 25

log6 2+ log6 3 = 1

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

= log6(2i3)

= log6(6)

6x = 6

x = 1

= log5

2008( )

= log5(25)

5x = 25

x = 2

log6 2+ log6 3 = 1

Example 1Simplify without a calculator. In other words, let’s use

what we know about logarithms!

a. log6 2+ log6 3 b. log5 200− log5 8

= log6(2i3)

= log6(6)

6x = 6

x = 1

= log5

2008( )

= log5(25)

5x = 25

x = 2

log6 2+ log6 3 = 1 log5 200− log5 8 = 2

Example 1

c. log85− log17 + 12

log400

Example 1

c. log85− log17 + 12

log400

= log 85

17( )+ 12

log400

Example 1

c. log85− log17 + 12

log400

= log 85

17( )+ 12

log400

= log5+ 12

log400

Example 1

c. log85− log17 + 12

log400

= log 85

17( )+ 12

log400

= log5+ 12

log400

= log5+ log40012

Example 1

c. log85− log17 + 12

log400

= log 85

17( )+ 12

log400

= log5+ 12

log400

= log5+ log40012

= log5+ log20

Example 1

c. log85− log17 + 12

log400

= log 85

17( )+ 12

log400

= log5+ 12

log400

= log5+ log40012

= log5+ log20

= log(5i20)

Example 1

c. log85− log17 + 12

log400

= log 85

17( )+ 12

log400

= log5+ 12

log400

= log5+ log40012

= log5+ log20

= log(5i20)

= log100

Example 1

c. log85− log17 + 12

log400

= log 85

17( )+ 12

log400

= log5+ 12

log400

= log5+ log40012

= log5+ log20

= log(5i20)

= log100 = 2

Example 2Rewrite without logarithms.

a. ln x = 1

3lna b. log x = loga − 4 logb

Example 2Rewrite without logarithms.

a. ln x = 1

3lna b. log x = loga − 4 logb

ln x = lna13

Example 2Rewrite without logarithms.

a. ln x = 1

3lna b. log x = loga − 4 logb

ln x = lna13

x = a13

Example 2Rewrite without logarithms.

a. ln x = 1

3lna b. log x = loga − 4 logb

ln x = lna13

x = a13

log x = loga − logb4

Example 2Rewrite without logarithms.

a. ln x = 1

3lna b. log x = loga − 4 logb

ln x = lna13

x = a13

log x = loga − logb4

log x = log a

b4( )

Example 2Rewrite without logarithms.

a. ln x = 1

3lna b. log x = loga − 4 logb

ln x = lna13

x = a13

log x = loga − logb4

log x = log a

b4( ) x = a

b4

Example 3Write an equation without exponents equivalent to the

exponential equation A = Pert.

Example 3Write an equation without exponents equivalent to the

exponential equation A = Pert.

A = Pert

Example 3Write an equation without exponents equivalent to the

exponential equation A = Pert.

A = Pert

P P

Example 3Write an equation without exponents equivalent to the

exponential equation A = Pert.

A = Pert

P P

AP= ert

Example 3Write an equation without exponents equivalent to the

exponential equation A = Pert.

A = Pert

P P

AP= ert

lnAP= lnert

Example 3Write an equation without exponents equivalent to the

exponential equation A = Pert.

A = Pert

P P

AP= ert

lnAP= lnert

ln A− lnP = rt

Example 4Which number is larger, 400500 or 500400?

Example 4Which number is larger, 400500 or 500400?

Let x = 400500 and y = 500400.

Example 4Which number is larger, 400500 or 500400?

Let x = 400500 and y = 500400.

log x = log400500

Example 4Which number is larger, 400500 or 500400?

Let x = 400500 and y = 500400.

log x = log400500 = 500log400

Example 4Which number is larger, 400500 or 500400?

Let x = 400500 and y = 500400.

log x = log400500 = 500log400 ≈ 1301.029996

Example 4Which number is larger, 400500 or 500400?

Let x = 400500 and y = 500400.

log x = log400500 = 500log400 ≈ 1301.029996

log y = log500400

Example 4Which number is larger, 400500 or 500400?

Let x = 400500 and y = 500400.

log x = log400500 = 500log400 ≈ 1301.029996

log y = log500400 = 400log500

Example 4Which number is larger, 400500 or 500400?

Let x = 400500 and y = 500400.

log x = log400500 = 500log400 ≈ 1301.029996

log y = log500400 = 400log500 ≈ 1079.588002

Example 4Which number is larger, 400500 or 500400?

Let x = 400500 and y = 500400.

log x = log400500 = 500log400 ≈ 1301.029996

log y = log500400 = 400log500 ≈ 1079.588002

So 400500 is larger.

Homework

Homework

p. 401 #1-25