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Unformatted text preview: Name___________________________________ 3.5: Exponential and Logarithmic Equations
SOLVE EXPONENTIAL EQUATIONS
Equality of Exponents Theorem
x y If b = b , then x = y, provided that b > 0 and b 1. Example: Solve for x algebraically. 3 x  2 = 81
Solution: 3 x  2 x  2 = 81 4 Notice that 81 is a power of 3. Write each side as a power of 3. Equate the exponents.
Check: 3 3
x  2 6  2 4 3 = 3 x  2 = 4 x = 6 81 81 81 81 3 81 Example: Solve for x, algebraically. 7 = 63
Solution: x log 7 x 7 = 63 = log 63 x Notice that 63 is not a power of 7. Take the logarithm of each side. (Its convenient to use either log or ln.) x log 7 = log 63 x =  log 63 log 7 x 2.129 Use the power property of logs. Exact solution Approximate solution Example: Solve for x algebraically. 2 2x  3 = 5 x  1 Example: Solve for x, algebraically. ex + 1 = 20 SOLVE LOGARITHMIC EQUATIONS
Example: Solve for x algebraically. log2 2x  3 = log2 x + 4
Solution: log2 2x  3 = log2 x + 4 log2 2x  3 = log2 x + 4 2x  3 = x + 4 x = 7 Cancel logarithms by the onetoone property. Solve for x. Check: log 2x  3 2 log 2 7  3 2 log 14  3 2 log 11 2 log x + 4 2 log 7 + 4 2 log 11 2 log 11 2 Example: Solve for x, algebraically. log 9x + 1 = 3
Solution: log 9x + 1 = 3 log10 9x + 1 = 3 Notice the base of the common log is 10. Convert the log equation to exponential form. Solve for x. 9x + 1 = 103 9x + 1 = 1000 9x = 999 x = 111 Check: log 9x + 1 log 9 111 + 1 log 999 + 1 log 1000 3 3 3 3 3 3 Example: Solve for x algebraically. log x + log x + 15 = 2 Example: Solve for x, algebraically. 1 + log 3x  1 = log 2x + 1 Example: Solve for x, algebraically. ex  ex = 6 2 Example: Solve for x, algebraically. 10x  10x 1 = 10x + 10x 2 ...
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 Spring '08
 MatRoy
 Logarithmic Equations, Equations, Exponents

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