3.5 Exponential and Logarithmic Equations

3.5 Exponential and Logarithmic Equations -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online