9-2-3b - 2-3 Part II Solving Exponential and Logarithmic...

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Unformatted text preview: 2-3 Part II: Solving Exponential and Logarithmic Equations Recall: log and exponent functions are inverses; that is: if you if you • start with the number 5 • raise e to the 5th power • then take the loge (ln) of the result • you just end up with the original 5 • start with the number 5 • take the loge (ln) of it • then raise e to that power • you just end up with the original 5 • try it on your calculator! Mathematically, ln e5 = 5 and eln 5 = 5. In general, we have the inverse properties of exponentiation and logs: ln ex = x and eln x = x These properties are indispensable for the methods to be shown. 2-3 Part II p. 1 How to solve exponential equations • • • how do you solve equations like: e3x - 2 = 5 ? methods we know won't work, because x is in an exponent! how can we get x out of the exponent, so we can solve as a non-exponential equation? The idea: Isolate exponential and use logs Example: Solve for x: e 3x - 2 = 5 1. take ln of each side (base is e) 2. apply inverse property ln e3x - 2 = ln 5 3x - 2 = ln 5 note: steps 1 and 2 can be shortcut into one step 3. solve Example: x = (ln 5 + 2)/3 = 1.203 Solve I = 1 - e-Rt/L 1. isolate exponential (for t) e-Rt/L = 1 - I 2. take logs/use inverse property -Rt/L = ln (1 - I) 3. solve 2-3 Part II t = -L ln(1 - I)/R p. 2 How to solve logarithmic equations • how do you solve equations like: ln x = (2/3)ln 8 + (1/2)ln 9 ? • methods we know won't work, because x is in an argument of a log! • how can we get x out of the ln, so we can solve as a nonlogarithmic equation? The idea: Use contraction, then one-to-one property or exponentiation! Example: ln x = (2/3)ln 8 + (1/2)ln 9 contract right-hand side ln x = ln 12 use one-to-one (eliminate ln) Example: contract x = 12 log (x + 3) - log x = 1 log((x+3)/x) = 1 exponentiate (raise 10 to lhs and rhs of equation) (x+3)/x = 10 solve x = 1/3 2-3 Part II p. 3 ...
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9-2-3b - 2-3 Part II Solving Exponential and Logarithmic...

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