Properties of Logarithms

Product property of logarithmsFor all positive numbers m, n, andb, where b 1,logbmn = logbm + logbn.

Example1. Given log35 1.4650,find each logarithm.

a. log345

log3(9•5)

log39 + log35

2 + 1.4650 = 3.4650

Example1. Given log35 1.4650,find each logarithm.

b. log325

log3(5•5)

log35 + log35

1.4650 + 1.4650 = 2.9300

Quotient property of logarithmsFor all positive numbers m, n, andb, where b 1,logbm/n = logbm - logbn.

Example2. Given log45 1.1610 and log415 1.9534, find eachlogarithm.

a. log45/16

log45 - log416

1.1610 - 2 = -0.8390

Example2. Given log45 1.1610 and log415 1.9534, find eachlogarithm. b. log43How can I rewrite 3 using 5 and 15?

log415/5log415 - log45

1.9534 - 1.1610 = 0.7924

Example3. The pH of a substance isthe concentration of hydrogen ions,[H+], measured in moles of hydrogen per liter of substance.

It is given by the formula,pH = log10(1/[H+])

Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2.

Example3. The pH of acid rain.It is given by the formula,pH = log10(1/[H+])

Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2.4.2 = log10(1/[H+])

4.2 = log101 - log10[H+]

Example3. The pH of acid rain.Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2.

4.2 = log10(1/[H+])

4.2 = log101 - log10[H+]

4.2 = 0 - log10[H+]

4.2 = -log10[H+]

Example3. The pH of acid rain.4.2 = log101 - log10[H+]

4.2 = 0 - log10[H+]

4.2 = -log10[H+]

-4.2 = log10[H+]

10-4.2 = [H+]

[H+] = 10-4.2 0.000063

Power property of logarithmsFor any real number p and positive numbers m, and b, where b 1,logbmp = p•logbm .

Example4. Solve

a. 2log36 - (1/4)log316 = log3x

b. log10z + log10(z+3) = 1

Example4. Solve

a. 2log36 - (1/4)log316 = log3x

log362 - log3161/4 = log3x

log336 - log32 = log3x

log336/2 = log3x

log318 = log3x x = 18

Example4. Solve

b. log10z + log10(z+3) = 1

log10z(z+3) = 1

z(z+3) = 101

z2 + 3z - 10 = 0

(z+5)(z-2) = 0z = -5 or z = 2

z = 2 is thesolution.

Example5. Rewrite as one logarithm

log102 + log10(x+9) + log10(y+6)

The properties allow us to rewritethese two additions as a singlemultiplication problem.

log10[2(x+9)(y+6)]

log10(2xy+12x+18y+108)

Example6. Rewrite as an equivalentlogarithmic expression

loga √4769 loga

4769

12

loga

4769

12

loga47-loga6912

loga47 - loga6912

12

Example7. Rewrite as an equivalentlogarithmic expression

loga

x4y9

z5

16

loga

x4y9

z516

logax4y9-logaz516

loga √x4y9

z56

logax4 + logay9-logaz516

Example7. Rewrite as an equivalentlogarithmic expression

logax4 + logay9-logaz516

4logax + 9logay-5logaz16

logax + logay- logaz23

32

56