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Math 1310
Section 5.4
Laws of Logarithms
The laws:
YOU MUST LEARN THESE!
If m, n and b are positive numbers, b ≠ 1 then
1. log b = 1
b
2. log 1 = 0
b
3. log (PQ) = log P + log Q
b
b
b
⎛P⎞
4. logb ⎜ ⎟ = logb P − logb Q
⎝Q⎠
1
Note: logb = − logb Q Why?
Q
5. log Pn = n log P
b
b
6. blogb P = P
7. log bP = P
b
8.
(Change of base formula)
logb x
loga x =
logb a
Note: this results in
ln x .
loga x =
ln a
Examples using these properties:
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Errors to avoid:
loga (x + y) ≠ loga x + loga y
loga x
≠ loga x − loga y
loga y
(loga x)3 ≠ 3 loga x
Using the properties to simplify logarithms
Example 1: Simplify each logarithm
log2 (32)
log6 9 + log6 4
log4 ( 96 ) − log4 (6)
⎛ 1⎞
log3 ⎜ ⎟
⎝ 81 ⎠
( )
log2 46
logb
( b)
( )
⎛ 1 ⎞
loga ⎜ ⎟ − loga a2
⎝ a3 ⎠
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Separating One Complicated Logarithmic Expression:
Example 2: Rewrite the expression in a form with no logarithm of a
product, power, or quotient.
log3 (x(x + 4))
Example 3: Rewrite the expression in a form with no logarithm of a
product, power, or quotient.
log
3
x
log ( 3x )
Example 4: Rewrite the expression in a form with no logarithm of a
product, power, or quotient.
⎛ ab3 ⎞
ln ⎜
⎟
⎜ c2 d ⎟
⎝
⎠
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Example 5: Rewrite the expression in a form with no logarithm of a
product, power, or quotient.
⎛ x+5 ⎞
log3 ⎜
⎟
⎝ x2 − 4 ⎠
Example 6: Rewrite the expression in a form with no logarithm of a
product, power, or quotient.
⎡ x2 ( x + 1 ) ⎤
⎥
log10 ⎢
4
⎢ ( x − 3 )( x + 7 ) ⎥
⎣
⎦
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Example 7: Rewrite the expression in a form with no logarithm of a
product, power, or quotient.
⎛ x +1⎞
log7 ⎜
3 ⎟
⎝ x
⎠
Example 8: Rewrite the following so that each logarithm contains a
prime number.
log2 35
log3 100
Combining a sum of logarithmic expressions:
Example 9: Rewrite as a single logarithm.
log3 x + log3 2
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Example 10: Rewrite as a single logarithm.
(
)
log x2 − 16 − log ( x + 4 )
Example 11: Rewrite as a single logarithm.
2 ln x − 5 ln(x + 1) + 1 ln(x − 3)
2
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Example 12: Rewrite as a single logarithm.
3 log5 (x + 2) − 2 log5 (x − 1) − 2 log5 (x − 7 )
Example 13: Rewrite as a single logarithm.
( ) ( ) ( )
ln A5 + ln A3 − ln A6
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The change of base formula:
Why? The change of base formula is typically used in three situations
- When an expression or equation involves logs in two or more
different bases
- When we want to evaluate a logarithm on a calculator and need
to convert a base to base 10 or base “e” to use our calculator.
- Many more advanced formulas in math are given in terms of
the natural logarithm, so if our equation is in a different base,
we need to change it to the natural logarithm to use the formula.
Example 14: Use the change of base formula to change log2 17 to
natural log.
Example 15: Use the change of base formula to change log7 12 to
base 10.