Properties of Logarithms Change of Base Formula:.
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Transcript of Properties of Logarithms Change of Base Formula:.
Properties of Logarithms
log 1a loga a log ra a loga ra
Product Rule: log ( ) log loga a aMN M N
Quotient Rule: log log loga a aM
M NN
Power Rule: log logra aM r M
log lnlog
log lnaM M
Ma a
Change of Base Formula:
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
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-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
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-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
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f (x) ax
The inverse function of an Exponential functionsis a log function.
f 1(x) loga x
Domain:Range:Key Points:Asymptotes:
Graphing Logarithmic Functions
xaxalog
Section 4.5Properties of Logarithms
Condense and Expand Logarithmic Expressions.
Rewrite expression to get same base on each side of equal sign.
where u and v are expressions in x
where u and v are expressions in x
If au av, then uv
822
1 2) x
xx 23 33)1
Type 1. Solving Exponential Equations
32164
1 3) x
Exponential Equations with base eTreat as a number.
3
2 1
2
eee xx
vuee v then ,ue
xx
xx ee
e
1
56 )(e 4)2 ee xx
Rewrite these expressions to have a single base e on both sides of the equation
Type 2 Solving: Log = Log
If then u = v vu aa loglog
)(log2log 5) 233 xx
When solving log functions, we must check that a solution lies in the domain!
)64ln(13ln( 6) x)x
Type 3. Solving: Log ( ) = Constant• Isolate and rewrite as exponential
3)12(og 7) 2 xl
2)7(log6-32 8) 23 x
4)6(log4 9) 27 xx
)1ln( 10-8007 10) t
Type 4: Exponential = Constant
Isolate exponential part and rewrite as log
210 )11 3 m
55 12) 1 xe
9)21(4 :one Try this 1 x
1. Power Rule“Expanding a logarithmic expression”Rewrite using the power rule.
15 )3(og 2) xl
)ln( 1) x
2. Product Rule“Expanding a logarithmic expression”
Rewrite using the Product Rule.
))4(ln( 2) 32 xe
3)1)(4(og 3) xxl
)3(og 1) 45 xl
3. Quotient Rule“Expanding a logarithmic expression”
Rewrite using the Quotient Rule.
24
16og 1)
xl
4. Expand the following expressions completely
1 23
2( 2)
2) ln1
x
x
x
x
25log 1) 2
5. Condensing Logarithmic Expressions
Rewrite as a single log expression
32log2og 1) 44 l
)log(3-4xog 2) xl
1)ln(x4
1ln(x)2 3)
Coefficients of logarithms must be 1 before you can condense them.
233 3 32) 15log log 9 log 9x x
2xlog3
1-1)log(2x4log(x)
2
1 1)
More practice….
7. Change-of-Base Formula
log lnlog
log lnaM M
Ma a
Example.
Find an approximation for )5(log2
