How do ratios of logs in different bases compare?

1. What is $\frac { \operatorname { log } _ { 2 } 32 } { \operatorname { log } _ { 2 } 4 }$?

   Hint (a:)

   Use your calculator to evaluate this expression.

   Answer (a):

   $\frac{5}{2} \text{ or }\ 2.5$

2. What is $\frac { \operatorname { log } 32 } { \operatorname { log } 4 }$?

3. What do you notice about your answers for parts (a) and (b)? Use the change of base formula to explain your results.

4. Change $\text{log}_{2}\left(7\right)$ into a logarithmic expression using a base of $5$.

   Step 1:

   Start with the definition of the logarithm.
   $\text{log}_{2}7 = x$
   $2^{x} = 7$

   Step 2:

   Take the log base $5$ of both sides.
   $\text{log}_{5}(2x) = \text{log}_{5}7$

   Step 3:

   Use the Power Property of Logarithms.
   $x\left(\text{log}_{5}2\right) = \text{log}_{5}7$

   Answer:

   $\log_{2}7 =\frac{\log_{5}7}{\log_{5}2}$

   Notice that this answer shows a very quick way to change bases.