5.5 Properties and Laws of Logarithms

xeln 4. x 10log 3.

x 3log 2. x 64log 1.

126

274

Do Now: Solve for x.

x = 3

x = 12x = 6

x = 1/3

Consider some more examples…

Without evaluating log (678), we know the expression “means” the exponent to which 10 must be raised in order to produce 678.

log (678) = x 10x = 678

If 10x = 678, what should x be in order to produce 678?

x = log(678) because 10log(678) = 678

And with natural logarithms…

Without evaluating ln (54), we know the expression “means” the exponent to which e must be raised in order to produce 54.

If ex = 54, what should x be in order to produce 54?

x = ln(54) because eln(54) = 54

ln (54) = x ex = 54

Basic Properties of Logarithms

Common Logarithms Natural Logarithms

1. log v is defined only when v > 0.

1. ln v is defined only when v > 0.

2. log 1 = 0 and log 10 = 1

2. ln 1 = 0 and ln e = 1

3. log 10k = k for every real number k.

3. ln ek = k for every real number k.

4. 10logv=v for every v > 0.

4. elnv=v for every v > 0.

** NOTE: These properties hold for all bases –

not just 10 and e! **

Example 1: Solving Equations Using Properties

Use the basic properties of logarithms to solve each equation.

71)ln(2x 2. 53)log(x 1.

100,003 x

310 x

103 x

1010

5

5

53)log(x

21e

7

7

71)ln(2x

7

x

1e2x

e12x

ee

Laws of LogarithmsBecause logarithms represent exponents, it is

helpful to review laws of exponents before exploring laws of logarithms.

When multiplying like bases, add the exponents.

aman=am+n

When dividing like bases, subtract the exponents. nm

n

m

aaa

Product and Quotient Laws of Logarithms

For all v,w>0, log(vw) = log v + log

wln(vw) = ln v + ln w

w ln v ln ln

w log v log log

0,w v,allFor

wv

wv

Using Product and Quotient Laws

1. Given that log 3 = 0.4771 and log 4 = 0.6021, find log 12.

2. Given that log 40 = 1.6021 and log 8 = 0.9031, find log 5.

log 12 = log (3•4) = log 3 + log 4 = 1.0792

log 5 = log (40 / 8) = log 40 – log 8 = 0.6990

Power Law of Logarithms

For all k and v > 0, log vk = k log v

ln vk = k ln v

For example…

log 9 = log 32 = 2 log 3

Using the Power Law1. Given that log 25 = 1.3979, find log .

2. Given that ln 22 = 3.0910, find ln 22.

4 25

log (25¼) = ¼ log 25 = 0.3495

ln (22½) = ½ ln 22 = 1.5455

Simplifying Expressions

Logarithmic expressions can be simplified using logarithmic properties and laws.

Example 1: Write ln(3x) + 4ln(x) – ln(3xy) as a single logarithm.

ln(3x) + 4ln(x) – ln(3xy) = ln(3x) + ln(x4) – ln(3xy)

= ln(3x•x4) – ln(3xy)

= ln(3x5) – ln(3xy)

=

=

3xy3x5

ln

yx4

ln

Simplifying Expressions

Simplify each expression.

1. log 8x + 3 log x – log 2x2

2. 4 2xx exlnln ¼

log 4x2