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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
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