7.5 Apply Properties of Logarithms

PROPERTIES OF LOGARITHMS

Let b, m, and n be positive numbers such that b ¹ 1

Product Property log_b mn = log_b m _+__ log_b __n__

Quotient Property log_b = log_b m __-___ log_b _ n __

Power Property logb mⁿ = __n log_b m_

Example 1

Use log₅ 4 » 0.861 and log₅ 9 » 1.365 to evaluate the logarithm.

a. log5 = log5 4 __-__ log5 9 » _0.861 - 1.365__ = _-0.504_	Quotient property Use given values. Simplify.
b.log5 36 = log5 (4 · 9) = log5 4_+_ log5 9 » _0.861 + 1.365__ = _2.226_	Write 36 as _4 · 9_. Product property Use given values. Simplify.
c. log5 81 = log5__9^2_ = _2 log5 9_ » __2(1.365)__ = __2.73_	Write 81 as __9^2__ Power property Use given value. Simplify.

Example 2

Expand log_{3 .}

log₃ = __log₃7x² - log₃ y__ Quotient property

= _log₃ 7 + log₃ x² - log₃y__ Product property

= _log₃ 7 + 2log₃x- log₃ y_ Power property

When you are expanding or condensing an expression involving logarithms, you may assume any variables are positive.

Example 3

Condense a logarithmic expression

Condense log 2 + 3 log 3 - log 9.

log 2 + 3 log 3 - log 9

= __log 2 + log3³ - log 9 Power property

= _log (2 · 3³)__ - log 9 Product property

= log __________ Quotient property

= _log 6_ Simplify.

CHANGE OF BASE FORMULA

If a, b, and c are positive numbers with b ¹ 1 and c ¹ 1, then

log_c a =

In particular log_c a = and log_c a =

Example 4

Use the change-of-base formula

Evaluate log₆ 11 using common logarithms and natural logarithms.

Solution

Using common logarithms:

log₆ 11 = » _______ » __1.3383__

Using natural logarithms:

log₆ 11 = » _______ » _1.3383__