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381 Rules for Logarithms

381 Rules for Logarithms - 1 ln ln by using the chain rule...

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Rules for Logarithms 1. ) ln( ) ln( ) ln( b a b a + = 2. ) ln( ) ln( ln b a b a - = 3. ) ln( ) ln( a x a x = 4. 0 ) 1 ln( = 5. [ ] x dx x d 1 ) ln( = By chain rule, [ ] dx dU U dx dU dU x U d dx x U d 1 * ) ( ln ) ( ln = = . Using our rules about logarithms we can calculate growth rates. Let’s take for example the quantity equation: MV = PY . Taking the natural log of both sides yields: Y P V M PY MV ln ln ln ln ln ln + = + = by our first rule of logarithms. To take the growth rate note that M, V, P, and Y are now functions of time ( t ). ) ( ln ) ( ln ) ( ln ) ( ln t Y t P t V t M + = + To find the growth rate, we are simply going to take the derivative with respect to time. dt t Y d dt t P d dt t V d dt t M d ) ( ln ) ( ln ) ( ln ) ( ln + = + Since all of those terms are similar, lets focus on the first one. dt M dt M M dt t M t dM dt t dM t M dt t dM t dM t M d dt t M d = = = = =
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Unformatted text preview: % ) ( ) ( ) ( * ) ( 1 ) ( * ) ( ) ( ln ) ( ln by using the chain rule we have calculated the percentage growth rate of M over time. Continuing with the rest of the equation yields: Y P V M ∆ + ∆ = ∆ + ∆ % % % % From taking the derivative of natural logs, we can derive the following rules about growth rates: Growth Rate Rules 1. Growth Rate of ( a*b ) = Growth Rate ( a ) + Growth Rate ( b ) 2. Growth Rate of b a = Growth Rate ( a ) - Growth Rate ( b ) 3. Growth Rate of b a = b * Growth Rate ( a )...
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