The Change of Base Formula
Next, we’ll consider how to modify the base of a formula. In data science, we commonly use base 10 or base 2. Sometimes we use natural logarithms (NI), which is a logarithm to the base e. We will discuss this later. Regardless of base, it’s possible to convert an equation from one base to another using a simple formula. If we want to take the logarithm of 12 to the base 2, we can use our pocket calculator or Excel spreadsheet, or computer and get 3.585. If we want to take the logarithm of 12 to the base 10, we’ll get 1.079-a very important point to keep in mind: that you’re not gonna get the same answer with a different base. The logarithm to the base 10 of 7 is more than one-third less than that of 2. Likewise. if we take logarithms to base 2 of 7 and 7, we get 2.807 and 0.8451 respectively. What do we have here? The numbers 2 to the 3.585 and 10 to the 1.079 both equal 12. Also, 2 to the 2.807 and 10 to 0.8451 both equal 7. If we have a new base a, and our original base is x, we can take our original formula which is log to the base x of b. We divide by the logarithm of the old base to the new base. This means we can work an example and illustrate what I mean. We have log to the base 10 of 12. We want to convert this to base 2, so we divide by log to the base 10 of 2. Log to the base 10 of 2 = 0.30103. So we have 1.079 divided by 0.30103 and that equals 3.585, which is log to the base 2 of 12. To convert log 2 of 7 into log 10 of 7, divide by log to the base 2 of 10. The base-2 logarithm of 10 is 3.3219, so 2.8073 divided by that number equals 0.8540. To convert one base to another, divide the original base by the logarithm of the original base over the new base. Practice problems help to familiarize you with this concept and it becomes more intuitive with time. Change of base log_2(12)=3.585 log_2(7)=2.807 log_10(12)=1.079 log_10(7)=.8451 2^3.585=12 2^2.807 10^1.079=12 10^.8451=7 Old Base is X New Base is a log_a(b)=logx(b)/log_x(a) New log(12)=1.079 log_10(2)=1.079/3.0103 Base a=2 log_2(12)=3.585 New log(7)=2.8073 log_2(10)=2.8073/3.3219 Base a=10 =.8540
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