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Unformatted text preview: 23 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. 23 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
nonexponential 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  eRt/L 1. isolate exponential (for t)
eRt/L = 1  I 2. take logs/use inverse property Rt/L = ln (1  I)
3. solve 23 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 onetoone property or
exponentiation!
Example: ln x = (2/3)ln 8 + (1/2)ln 9
contract righthand side
ln x = ln 12
use onetoone (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 23 Part II p. 3 ...
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 Spring '11
 Staff
 Equations, Exponentiation, Inverse function, Logarithm

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