Find the approximate value of log 20 to the base four, given that log five to the base four is approximately equal to 1.161.
In order to answer this question, we need to recall one of our laws of logarithms. Log 𝑎 plus log 𝑏 is equal to log of 𝑎𝑏. This law is only true if our base number is the same. As 20 is equal to four multiplied by five, we could write log 20 as log four multiplied by five. Using the law that we quoted, we could rewrite this as log four to the base four plus log five to the base four.
We are told in the question that log five to the base four is equal to 1.161. One of our other important facts about logarithms is that log 𝑎 to the base 𝑎 is equal to one. This is because 𝑎 to the power of one is equal to 𝑎. Log four to the base four will, therefore, be equal to one. Adding 1.161 to one gives us 2.161. Log 20 to the base four is approximately equal to 2.161.