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# 8.4 properties of logarithms

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### 8.4 properties of logarithms

1. 1. 8.4 – Properties of Logarithms
2. 2. Properties of Logarithms  There are four basic properties of logarithms that we will be working with. For every case, the base of the logarithm can not be equal to 1 and the values must all be positive (no negatives in logs)
3. 3. Product Rule logbMN = LogbM + logbN  Ex: logbxy = logbx + logby  Ex: log6 = log 2 + log 3  Ex: log39b = log39 + log3b
4. 4. Quotient Rule M log b = log b M − log b N N  x Ex: log 5 = log 5 x − log 5 y y  a Ex: log 2 = log 2 a − log 2 5 5  MN = log 2 M + log 2 N − log 2 P Ex: log 2 P
5. 5. Power Rule log b M = x log b M x    Ex: log 5 B = 2 log 5 B Ex: log 2 5 = x log 2 5 Ex: 2 x log 7 a b = 3 log 7 a + 4 log 7 b 3 4
6. 6. Let’s try some  Working backwards now: write the following as a single logarithm. log 4 4 − log 4 16 log 5 + log 2 2 log 2 m − 4 log 2 n
7. 7. Let’s try some  Write the following as a single logarithm. log 4 4 − log 4 16 log 5 + log 2 2 log 2 m − 4 log 2 n
8. 8. Let’s try something more complicated . . . Condense the logs log 5 + log x – log 3 + 4log 5 log4 5 − 2 log4 x + 5(log4 3x − log4 5x)
9. 9. Let’s try something more complicated . . . Condense the logs log 5 + log x – log 3 + 4log 5 log4 5 − 2 log4 x + 5(log4 3x − log4 5x)
10. 10. Let’s try something more complicated . . .  Expand 4 10 x log 3y2 2 x   log8   5    3
11. 11. Let’s try something more complicated . . .  Expand 4 10 x log 3y2 2 x   log8   5    3