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College Algebra 5.4

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College Algebra 5.4

1. 1. 5.4 Exponential and Logarithmic Equations <ul><li>One-to-one and inverse properties </li></ul><ul><li>Solve exponential equations </li></ul><ul><li>Solve log equations </li></ul>
2. 2. A. One-to-one and inverse properties <ul><li>The one-to-one property in this section is just a little different from before. </li></ul><ul><li>log a x = log a y would imply that x = y. </li></ul><ul><li>lnx – ln(x + 2) = ln(x + 6) </li></ul>
3. 3. Inverse Property: <ul><li>Remember that neato product property of logs? log 3 x = </li></ul><ul><li>What power! We can bring exponents down! So what if we had an exponential equation like: 3 x = 14 </li></ul><ul><li>We cannot use the one-to-one property on it because we cannot write 14 as an expression with base 3. If only…. </li></ul>
4. 4. 3 x = 14 <ul><li>If only there were a “log” or an “ln” there…. Then we could bring down the x…. </li></ul><ul><li>WE CAN TAKE THE “LOG” OR “LN” OF BOTH SIDES!! </li></ul><ul><li>ln 3 x = ln 14 Then product property. </li></ul><ul><li>x ln3 = ln 14 Then solve for x… </li></ul><ul><li>[Remember that things like “ln3” and “ln14” are just numbers.] </li></ul>
5. 5. B. Exponential Equations when you can’t solve using one-to-one <ul><li>STEP 1: Get the exponential expression by itself (if it isn’t already) </li></ul><ul><li>STEP 2: Take the log or ln of both sides. </li></ul><ul><li>STEP 3: Pull the exponent down. </li></ul><ul><li>STEP 4: Solve for x. </li></ul>
6. 9. C. Solve log Equations <ul><li>We’ve done each of these before. </li></ul><ul><li>IF there are more than one logarithmic expression on one side of the equal sign, then try condensing them to make it one. </li></ul><ul><li>IF there are two logs, one on each side of the equals, then try using the one-to-one property (objective A of this lesson). </li></ul><ul><li>IF there is only one log on only one side of the =, then convert it to exponential form. </li></ul>