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1140 L23 - f ± 1 and its domain and range Applications ex...

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Lecture 23: Section 3.4 Exponential and Logarithmic Equations Two basic strategies for solving exponential or logarithmic equations: 1. One-to-One Properties a x = a y if and only if log a x = log a y if and only if 2. Inverse Properties a log a x = log a a x = ex. Solve for x : 1) 3 8 x 2 9 x 1 = 648 2) log x = 2
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Solving Exponential Equations 1. Isolate the exponential expression on one side of the equation. 2. Take the logarithm on each side, then use the Properties of Logarithms to bring down the exponent. 3. Solve for the variable. ex. Solve for x : 1) 3 e 2 x = 5 2) 5 x 1 = 3 2 x
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3) 9 x + 3 x 6 = 0
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Solving Logarithmic Equations 1. Isolate the logarithmic term on one side of the equation; you may need to rst combine the logarithmic terms. 2. Write the equation in exponential form (or raise the base to each side of the equation). 3. Solve for the variable. NOTE: Always check your answers when solving logarithmic equations. ex. Solve for x : 1) 1 + log 2 (25 x ) = 4 2) log 4 x log 4 ( x 1) = 1 2
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ex. Find the domain of f ( x ) = ln( e x +2 1). ex.
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Unformatted text preview: f ± 1 and its domain and range. Applications ex. A small lake is stocked with a certain species of ±sh. The ±sh population is modeled by the function P = 10 1 + 4 e ± : 8 t where P is the number of ±sh in thousands and t is measured in years since the lake was stocked. 1) Find the ±sh population after 3 years. 2) After how many years will the ±sh population reach 5000 ±sh? Practice 1. Find the inverse of f ( x ) = log 3 ( x ± 4) + 2. Find the domain and range of f and its inverse. 2. 5 ± x 2 = 2 3. log( x + 2) + log( x ± 1) = 1 4. 3 log(4 x ) = 12 5. e 2 x ± e x ± 6 = 0 6. Find the intersection of the 2 funcitons y = 2 x and y = 4 3 ± x Answer: 1) f ± 1 ( x ) = 4 + 3 x ± 2 , Domain of f ± 1 = Range of f = ( ±1 ; 1 ); Range of f ± 1 = Domain of f = (4 ; 1 ) 2) x = ± ln 4 ln 5 3) x = 3 only 4) x = 2500 5) x = ln 3 6)(2 ; 4)...
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