8.5 – Using Properties of Logarithms

• Product Property:

• Product Property: logbmn

• Product Property: logbmn = logbm

• Product Property: logbmn = logbm + logbn

• Product Property: logbmn = logbm + logbn

Ex.

• Product Property: logbmn = logbm + logbn

Ex. log95x

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property:

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m

n

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm

n

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex.

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex. log9 x

9

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex. log9 x = log9x

9

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex. log9 x = log9x – log99

9

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex. log9 x = log9x – log99

9

• Power Property:

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex. log9 x = log9x – log99

9

• Power Property: logb mp

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex. log9 x = log9x – log99

9

• Power Property: logb mp = plogbm

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex. log9 x = log9x – log99

9

• Power Property: logb mp = plogbm

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex. log9 x = log9x – log99

9

• Power Property: logb mp = plogbm

Ex.

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex. log9 x = log9x – log99

9

• Power Property: logb mp = plogbm

Ex. log9x7

• Product Property: logbmn = logbm + logbn

Ex. log95x = log95 + log9x

• Quotient Property: logb m = logbm - logbn

n

Ex. log9 x = log9x – log99

9

• Power Property: logb mp = plogbm

Ex. log9x7 = 7log9x

Ex. 2 Solve the following equations.

a. 3 log5 x – log5 4 = log5 16

Ex. 2 Solve the following equations.

a. 3 log5 x – log5 4 = log5 16

log5 x3 – log5 4 = log5 16

Ex. 2 Solve the following equations.

a. 3 log5 x – log5 4 = log5 16

log5 x3 – log5 4 = log5 16

log5 x3 = log5 16

4

Ex. 2 Solve the following equations.

a. 3 log5 x – log5 4 = log5 16

log5 x3 – log5 4 = log5 16

log5 x3 = log5 16

4

Ex. 2 Solve the following equations.

a. 3 log5 x – log5 4 = log5 16

log5 x3 – log5 4 = log5 16

log5 x3 = log5 16

4

x3 = 16

4

Ex. 2 Solve the following equations.

a. 3 log5 x – log5 4 = log5 16

log5 x3 – log5 4 = log5 16

log5 x3 = log5 16

4

x3 = 16

4

x3 = 64

Ex. 2 Solve the following equations.

a. 3 log5 x – log5 4 = log5 16

log5 x3 – log5 4 = log5 16

log5 x3 = log5 16

4

x3 = 16

4

x3 = 64

x = 4

b. log4 x – log4 (x – 6) = 2

b. log4 x – log4 (x – 6) = 2

log4 x = 2

x – 6

b. log4 x – log4 (x – 6) = 2

log4 x = 2

x – 6

Change to exponential form!!!

b. log4 x – log4 (x – 6) = 2

log4 x = 2

x – 6

42 = x

x – 6

b. log4 x – log4 (x – 6) = 2

log4 x = 2

x – 6

42 = x

x – 6

16 = x

x – 6

b. log4 x – log4 (x – 6) = 2

log4 x = 2

x – 6

42 = x

x – 6

16 = x

1 x – 6

b. log4 x – log4 (x – 6) = 2

log4 x = 2

x – 6

42 = x

x – 6

16 = x

x – 6

16(x – 6) = x

b. log4 x – log4 (x – 6) = 2

log4 x = 2x – 6

42 = xx – 6

16 = xx – 6

16(x – 6) = x16x – 96 = x

b. log4 x – log4 (x – 6) = 2

log4 x = 2x – 6

42 = xx – 6

16 = xx – 6

16(x – 6) = x16x – 96 = x

15x = 96

b. log4 x – log4 (x – 6) = 2 log4 x = 2

x – 6 42 = x

x – 6 16 = x

x – 6 16(x – 6) = x16x – 96 = x

15x = 96 x = 96/15

b. log4 x – log4 (x – 6) = 2 log4 x = 2

x – 6 42 = x

x – 6 16 = x

x – 6 16(x – 6) = x16x – 96 = x

15x = 96 x = 96/15 x = 32/5