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### Section 6.2 : Logarithm Functions

12. Without using a calculator determine the exact value of $$\log \displaystyle \frac{1}{{100}}$$.

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Hint : Recall that converting a logarithm to exponential form can often help to evaluate these kinds of logarithms. Also recall what the base is for a common logarithm.
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Recalling that the base for a natural logarithm is 10 and converting the logarithm to exponential form gives,

$\log \frac{1}{{100}} = {\log _{10}}\frac{1}{{100}} = ?\hspace{0.25in} \Rightarrow \hspace{0.25in}{10^?} = \frac{1}{{100}}$ Show Step 2

Now, we know that if we raise an integer to a negative exponent we’ll get a fraction and so we must have a negative exponent and then we know that $${10^2} = 100$$. Therefore, we can see that $${10^{ - 2}} = \frac{1}{{100}}$$ and so we must have,

$\require{bbox} \bbox[2pt,border:1px solid black]{{\log \frac{1}{{100}} = - 2}}$