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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 bullet is shot upwards with an initial velocity of 100 ft/sec from a point 12 ft above the ground, and its height above the ground at time t is given by h(t)= -16t^2 + 100t +12
5(4a+1)+6=-(-11a+1)+6a
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Given f(x)= 2+3x-x 2 and g(x) =2x-1 evaluate ( fg ) ( x ) , (fog)(x) and (gof )(x) Solution These are the similar functions that we utilized in the first set of instances
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Bills Roast Beef sells 3 times as many sandwiches as Pete''s deli. The difference between their sales is 170 sandwiches. How many sandwiches did each sell?
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