Chapter6 - Pre-Calculus Chapter 6A Chapter 6A Exponential...

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© : Pre - Calculus - Chapter 6A Chapter 6A - Exponential and Logarithmic Equations Exponential Equations In previous chapters we learned about the exponential and logarithmic functions, studied some of their properties, and learned some of their applications. In this chapter we show how to solve some simple equations which contain the unknown either as an exponent (exponential equation) or as the argument of a logarithmic function. As a general rule of thumb , to solve an exponential equation proceed as follows : 1 . Isolate the expression containing the exponent on one side of the equation. 2 . Take the logarithm of both sides to ”bring down the exponent”. 3 . Solve for the variable. Example 1 : Solve 3 x 25 Solution: 3 x 25 take the natural log of both sides x ln3 ln25 solve for x x 2. 929947 Example 2 : Solve 4 3 x 1 8 Solution: 4 3 x 1 8 isolate x 3 x 1 4 take the natural log of both sides x 1 ln4 solve for x x 1 . 2618595 Example 3 : Solve the equation 10 1 e x 2 Solution We need to “isolate” the terms involving x on one side of the equation. We can do this by cross multilpying and then solving for e x : 1 e x 5 e x 4 x x ≈− 1. 386294 © : Pre - Calculus
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© : Pre - Calculus - Chapter 6A Example 4 : Solve the equation x 2 2 x 2 x 0. Solution: This looks slightly difficult. However, let’s factor the 2 x term out of the left hand side. x 2 2 x 2 x 0 2 x x 2 1 0 Since a product can equal zero if and only if one of the factors is zero, we know that if x is a solution, then either 2 x 0or x 2 1 0. But 2 x is never 0, thus, our solution must satisfy x 2 1 0 x 2 1 x  1 Example 5 : Solve the equation e 2 x 3 e x 2 0. Solution: This equation really looks hard, and it is until we notice that it is a quadratic equation in e x . To see that this is the case, set u e x , then the equation e 2 x 3 e x 2 0 can be written as u 2 3 u 2. Solving this latter equation we have e x 2 3 e x 2 0 u 2 3 u 2 0 u 1  u 2 0 Thus, we have u 1or u 2. In terms of e x , this means e x 1o r e x 2 x ln1 x ln2 x 0 x . 6931472 © : Pre - Calculus
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© : Pre - Calculus - Chapter 6A Logarithmic Equations In the previous page we showed how to solve some exponential equations. Here we solve some logarithmic equations. To solve a logarithmic equation proceed as follows 1 . Isolate the expression containing the logarithm on one side of the equation. 2 . Exponeniate both sides to remove the log function. 3 . Solve for the variable. Example 1 : Solve log x 35 for x . Solution: The main item we need to note here is that log represents the logarithm of a number to base 10. Thus, we need to raise both sides of the equation to the 10 th power.
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This note was uploaded on 02/23/2010 for the course CHEM 107 taught by Professor Generalchemforeng during the Fall '07 term at Texas A&M.

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Chapter6 - Pre-Calculus Chapter 6A Chapter 6A Exponential...

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