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$$ \log_3{100} - \log_3{18} - \log_3{50} $$

How do I solve?

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3 Answers

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$$\log_3{\frac{100}{18\cdot 50}}=\log_3{\frac{1}{9}}=-2$$

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Since they all have the same base ($3$), you can combine them. Remember that when logs are subtracted, you divide:

$$\log_3{100} - \log_3{18} - \log_3{50}$$ $$\log_3{\frac{100}{18}} - \log_3{50}$$ $$\log_3{\frac{\frac{100}{18}}{50}}$$ $$\log_3{\frac1{9}}$$

Remember the definition of logs - this is basically saying: to what power is 3 equal to $\frac1{9}$? The answer is -2 (negative makes it $\frac1{3}$, then the squared is so you get $9$).

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Notice that $$\log_3 100 - \log_3 18 - \log_3 50 = \log_3 \left(\frac{100}{18}\right) - \log_3 50 = \log_3 \left(\frac{100/18}{50}\right) = \log_3 \frac{1}{9} = -2.$$

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