sullivan algebra and trigonometry: section 6.5 properties of logarithms objectives of this section...
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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.
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