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Section 5.4 – Properties of Logarithmic Functions This section covers some properties of logarithmic function that are very similar to the rules for exponents. Properties of Logarithms For any positive number M and N, and any logarithmic base a, Product Rule: ( )log log loga a aM N M N⋅ = +
Quotient Rule: log log loga a aM M NN
⎛ ⎞ = −⎜ ⎟⎝ ⎠
Product Rule: log logpa aM p M= ⋅
Example 1: Express as a sum of logarithms by using the Product Rule.
( )3
3
9 27 (By the Product Rule)
(By the definition o
lo
g )
g
f lo
⋅ =
=
Example 2: Express as a single logarithm.
32 2 log (By the Produlog ct Rule)p q+ =
Example 3: Express 3log 11a
− as a product. Compare this to the left side of the Power Rule: log logp
a aM p M= ⋅ .
M = and p =
Now inserting these in to the right side of the power rule gives 3log 11 a
− = .
Express 4log 7a as a product.
First rewrite 4 7 as an exponent 1
using n nx x⎛ ⎞
=⎜ ⎟⎝ ⎠
:
4l 7 o g a = .
Then use the Power Rule: 4l og 7 a = .
2 Chapter 5 Exponential Functions and Logarithmic Functions
Example 8: Express as a single logarithm ( ) ( )2ln 3 1 ln 3 5 2
(By the Quotient Rule) (By factoring the denominator
)
x x x+ − − −
= =
=
(By canceling 3
1) x +
These properties of logarithms can also be used to find some unknown logarithm when given some particular logarithmic values. Example 9: Given that log 2 0.301a ≈ and log 3 0.477a ≈ , find log 6a if possible.
4 Chapter 5 Exponential Functions and Logarithmic Functions