Sullivan Algebra and Trigonometry: Section 6.5Properties of Logarithms

Objectives of this Section

• Work With the Properties of Logarithms

• Write a Log Expression as a Sum or Difference of Logarithms

• Write a Log Expression as a Single Logarithm

• Evaluate Logarithms Whose Base is Neither 10 nor e

Properties of Logarithms

loga a 1 a MaMlog loga

ra r

log log loga a aMN M N

log log loga a aMN

M N

log loga aNN

1

log logar

aM r M

1

3log

2

2

x

xxa

Write the following expression as the sum and/or difference of logarithms. Express all powers as factors.

log log loga a a

x x

xx x x

2

22 23

13 1

log log loga a ax x x2 23 1

212

3 2 1log log loga a ax x x

Write the following expression as a single logarithm.

1log12log4

1log3 xxx aaa

log log loga a ax x x3 142 1 1

log log loga a ax x x3 4 2 1 1

log loga ax x x3 4 2 1 1

logax xx

3 4 2 11

Theorem: If N = M, then NM aa loglog

Theorem: If , then N = M.NM aa loglog

These properties become critical when solving exponential and logarithmic equations, covered in the next section.

Most calculators only evaluate logarithmic functions with base 10 or base e. To evaluate logs with other bases, we use the change of base formula.

logloglogab

b

MMa

loglogMa

lnlnMa

Calculate log5 63

logloglog5 63

635

lnln

635

2 574.