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Section 10.6 Exponential and Log Equations To solve Exponential Equations you must first decide: Can you get the same base on both sides as we did before 2x 1 4 x 2 12 2 x 2 2 −2 x −2 or you cannot get the same base on both sides. 2x 7 If you cannot get the same base on both sides then we must use logarithms. First isolate the exponential part on one side alone 2 x 7 is already isolated 5 x − 3 13 is not isolated and you should move the 3 to right hand side 5 x 16 Next identify the base in your problem 10 x 500 base is 10 2 x−3 12 base is 2 Now we need a fact from logarithms if M N then log b M log b N 10 x3 45 log 10 10 x3 log 10 45 Recall another fact log b b x x log b M x x ∗ log b M Example 1 1 2 A Scientific Report 10 x3 45 log 10 10 x3 log 10 45 x 3 log 10 45 x −3 log 10 45 x −1. 346 8 Example 2 2x 7 ln2 x ln7 x ∗ ln2 ln7 ln2 x ln7 x 0. 356 21 Example 3 3e 2x 1 16 3e 2x 15 e 2x 5 lne 2x ln5 2x ln5 ln5 x 2 x 0. 804 If the base is 10 use log10, if the base is e then use "ln" and if the base is not "e" or "10" it does not matter what log you use. Change of Base You might have noticed that your calculator only has buttons for "log" and "ln" and you wondered what do you do if you needed log 5 125 Well have no fear we have a formula called Change of Base which allows you to use logs that your calculator does have. A Scientific Report 3 log b something log b a log 10 125 log 5 125 3 log 10 5 ln125 3 log 5 125 ln5 log a something) Logarithmic Equations These equations can be condensed into 2 types: Type 1 log b something number Type 2 log b something log b something else If your problem only has logs everywhere then it is type 2, but if it has a number anywhere in it, it is type 1. Solving Type 1 1. 2. Use your properties of logs to collapse the problem Use the basic fact that log b something number b number something 3. 4. Check all of the answers since the domain for the log is x 0. Please note that I am not saying throw away all negative answers. Suppose your problem has a piece like log 10 5 − x ... and you get answers of x 7, −2 then you would toss the 7 since it makes the INSIDE 0, but you would keep the -2 since 5 − −2 positive. Example Type 1 4 A Scientific Report log 2 5x − 3 5 2 5 5x − 3 32 5x − 3 35 5x x7 Example Type 1 log 10 x log 10 x 21 2 log 10 xx 21 2 10 2 xx 21 100 x 2 21x x 2 21x − 100 0 x 25x − 4 0 x −25, 4 x 4 and toss out the x −25 Solving Type 2 1. 2. Use your properties of logs to collapse the problem Use the following fact if log b M log b N then M N 3. Check all of the answers since the domain for the log is x 0. Example Type 2 ln2t 5 ln3 lnt − 1 ln2t 5 ln3t − 1 2t 5 3t − 3 8t