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Section 4: Solving Exponential and Logarithmic Equations SOLVING EQUATIONS WITH THE SAME BASE To solve an exponential equation, the following property is sometimes helpful: If a > 0, a ≠1, and ax = ay , then x = y . Similarly, we have the following property for logarithms: If log x = log y , then x = y . _______________________________________________________________________ Example: Solve )log (5x−6) = log (x+2 for x: 3 3 Solution: log3(5x−6)=log3(x+2) 5x−6=x+2 5x−x=2+6 4x=8 x = 2 _______________________________________________________________________ _______________________________________________________________________ Example: Solve 2x+1 =8for x. Solution: Here, the bases are not the same, but we find that we are able to manipulate the right hand side to make the bases the same. 2x+1 =8 2x+1 = 23 x+1=3 x = 2 _______________________________________________________________________ CANCELLATION LAWS In the “Introduction to Exponential and Logarithmic Functions” file, we learned that logs and exponentials are inverse functions. Since the exponential and logarithmic functions are inverse functions, cancellation laws apply to give: log (ax)= x for all real numbers x a aloga x = x for all x > 0 We already stated that e is the most convenient base to work with for exponential and logarithmic functions. The same cancellation laws apply for the natural exponential and the natural logarithm: ln(ex) = x for all real numbers x elnx = x for allx >0 These last two cancellation laws will be especially useful in calculus this year. To solve a simple exponential equation, you can take the natural logarithm of both sides. (technically, you can take the logarithm with any base, but the natural log is often the easiest). Similarly, to solve a simple logarithmic equation, you can take the natural exponential of both sides. At this point, the equation can be solved using basic algebra. _______________________________________________________________________ Example: Solve e2x = 8 for x. Solution: e2x = 8 ln(e2x) = ln(8) 2x = ln(8) x = ln(8) 2 _______________________________________________________________________ _______________________________________________________________________ Example: Solve ln(x + 5) = 4 for x. Solution: ln(x + 5) = 4 eln(x+5) = e4 x + 5 = e4 x = e4 −5 _______________________________________________________________________ For a detailed explanation of some more difficult examples, check out the mini-clips!
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