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Properties of Logarithms
1. logb1 = 0 . It follows from the fact that bo = 1.
2. logb b = 1. It follows from the fact that b 1= b .
3. logb bx = x . it can be generalized out to blog b f ( x ) = f ( x ).
4. b logb x = x . It can be generalized out to b logb f ( x ) = f ( x ) .
Properties 3 and 4 lead to a pleasant relationship among the logarithm & exponential function.
Let's first calculate the following function compositions for f ( x )= b x and g ( x ) = logb x .
( f o g )( x ) = f [g ( x )] = f (logb x ) = b logb x = x
( g o f ) ( x ) = g [f ( x )]= g [b x ] = log b bx = x
Remember again from the section on inverse functions which this means that the exponential & logarithm functions are inverses of each other. It is a nice fact to remember on occasion.
We have to also give the generalized version of Properties 3 & 4 in terms of both the natural and common logarithm
ln e f ( x ) = f ( x) log10 f ( x ) = f ( x)
eln f ( x ) = f ( x ) 10log f ( x ) = f ( x )
a^4b+a^2b^3
50 units for 580 dollars a month, rent increase 625.00 now only 47 units occupied
4f=48
a carpenter had a 20 ft piece of lumber which he cut into 3 pieces. one piece was 7 1/3 ft long and other was 6 1/2 ft long. how long was the remaining piece?
What is 789673+469767 equal to?
x^2+6x+8=0
3x+1=7
??2+??2+16??-18??+145=25 Standard form (x-h)^2 +(y-k)^2 k (x^2+16x+64)^2+(y^2-18y+81)^2=25 (x+8)^2+(y-9)^2=120 (h,k)=(8,-9) R=5 Intercepts
6x^3+3x^2-18x
Fact Following any system of equations there are accurately three possibilities for the solution. 1. There will not be a solution. 2. There will be just one solution.
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