3.4. Properties of Logs http://www.ck12.org
Example C
Simplify the following expression: 2 log 12144 −^4.
Solution:2 log 12144 −^4 =− 8 ·log 12122 =− 16 ·log 1212 =− 16
Concept Problem Revisited
A log expression represents an exponent. The expression log 101 ,000 represents the number 3. The reason to
keep this in mind is that it can solidify the properties of logs. For example, adding exponents implies bases are
multiplied. Thus adding logs means the bases of the exponents are multiplied.
Vocabulary
Alogarithmis a way of rewriting exponential equations to isolate the exponent.
Guided Practice
- Prove the following log identity:
logab=log^1 ba - Rewrite the following expression under a single log.
lne−ln 4x+ 2 (elnx·ln 5) - True or false:
(log 34 x)·(log 35 y) =log 3 ( 4 x+ 5 y)
Answers: - Start by letting the left side of the equation be equal tox. Then, rewrite in exponential form, manipulate, and
rewrite back in logarithmic form until you get the expression from the left side of the equation.
logab=x
bx=a
b=a^1 x
logba=^1 x
x=log^1
baTherefore,log^1 ba=logabbecause both expressions are equal tox.
- lne−ln 4x+ 2 (elnx·ln 5)
=ln(e
4 x)
+ 2 x·ln 5
=ln(e
4 x)
+ln( 52 x)
=ln(e· 52 x
4 x)
- Note, it may be very tempting to make errors in this practice problem.