Change of base: The final topic that we have to look at in this section is the change of base formula for logarithms. The change of base formula is,

                                        log _b x= log_a  x / log _a b

It is the most general change of base formula & will convert from base b to base a. Though ,the usual cause for using the change of base formula is to calculate the value of a logarithm that is in a base which you can't easily deal with.  By using the change of base formula means that you can write down the logarithm in terms of a logarithm which you can deal with. The two most common change of base formulas are

log_b x= ln x /ln b                     and       log_bx  =  log x/log b

Actually, often you will see one or the other listed as THE change of base formula!

           We can calculate the value of logarithms without the change of base formula since all the arguments could be written in terms of the base to a power. For example,

                                       log₇ 49 =2    because            7 ²= 49

Though, this only works since 49 can be written as a power of 7! We would required the change of base formula to compute log₇ 50 .

log ₇ 50= ln 50 / ln 7= 3.91202300543/1.94591014906  = 2.0103821378

OR

log ₇ 50= log 50 / log 7 = 1.69897000434/0.845098040014 = 2.0103821378

Hence, it doesn't matter that we use, we will get the similar answer regardless of the logarithm which we use in the change of base formula.

Note that we could employ the change of base formula on log₇ 49 if we desired to as well.

                   log ₇ 49= ln 49/ ln 7= 3.89182029811/1.94591014906 = 2

It is a lot of work though, and is probably not the best way to deal with this.

Hence, in this section we saw how logarithms work & took a look at some properties of logarithms.  We will run into logarithms on instance so ensure that you can deal with them while we do run into them.