3.5 Exponential and Logarithmic Equations

# 3.5 Exponential and Logarithmic Equations - Name 3.5...

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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 one-to-one 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 - e-x = 6 2 Example: Solve for x, algebraically. 10x - 10-x 1 = 10x + 10-x 2 ...
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