Section 1.8 : Logarithm Functions
6. Without using a calculator determine the exact value of \(\displaystyle \log \frac{1}{{100}}\).
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.
Recalling that the base for a common logarithm is 10 and converting the logarithm to exponential form gives,
\[\log \frac{1}{{100}} = {\log _{10}}\frac{1}{{100}} = ?\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}{10^?} = \frac{1}{{100}}\]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,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\log \frac{1}{{100}} = - 2}}\]