1. 7.5 Notes: Solving Logs

2. What if there is a “log” in our equation?  What if our equations already have a “log” or “ln” in them? Can we still add “log” or “ln” on the other side to solve?  No, we can’t. Remember, whatever we do to one side we have to do to the other. If we only add “log” or “ln” to one side, we aren’t following the rules.  There are three types of log equations we will encounter.  We MUST check for EXTRANEOUS SOLUTIONS!

3. Wait, extraneous solutions?  Yes, remember “extraneous” means we did the math correctly, but when we plug the answer back in to check our work it does not work.  When would this happen?  Let’s take a look at the log of a negative number:  log (-4)log (-1)log (-1/4)  So, what can’t happen?

4. Situation #1:  The logs and their bases on each side of the equation are EXACTLY the same. Then, we can treat these like we did the like bases. The property of equality says we can “ignore” the logs and just set the equation in parenthesis equal to each other to solve.  log 5 (4x – 7) = log 5 (x + 5)ln (7x – 4) = ln (2x + 11)

5. Situation #2  There is only one “log” in the equation. So, we use our “circle of logs” to rewrite the equation in exponential form to solve.  log 4 (5x – 11) = 3log 2 (x – 6) = 5

6. Situation #3  There are two logs ON THE SAME SIDE of the equal sign. We will use log properties to condense to one log, then rewrite as an exponential equation to solve.  Guess what….because we KNOW you missed it…. We get to FACTOR!!!!!  log 2x + log (x – 5) = 2log 4 (x + 12) + log 4 x = 3