So like in my Computer Science #4 post, I went over logarithms, but let’s refresh on exponents first.
In mathematical notation (the way of writing a number), exponents are a symbol that represents the repeated multiplication of a number by itself. If I want to write 3 x 3 x 3 x 3, without taking up space, I would write 34. If you’re multiplying the same number twice, to the second power, that’s known as ‘squared’. If you’re doing it three times, to the third power, that’s known as ‘cubed’. To the first power means the number is by itself, there’s no multiplication. Any number to the power of 0 is ALWAYS 1.
My 12th-grade pre-calc teacher told us that at her final math exam in college, the professor came in. You know those old-school chalkboards that take up the whole wall. He wrote an equation that took up the whole width of that board. Dozens of numbers. She said everybody was stressing trying to break down the equation. Some people were using pages of scratch paper. She recognized the equation started with an open parenthesis, and ended with a closing one, topped off with that tiny superscript 0 right after it. To that point, she wrote a simple 1 on her paper and turned it in. Her classmates thought she gave up. I don’t think she was the only one, but she was one of the few students of an upper-level college math class, to pass that final exam. No matter how many digits they throw at you, ANY number, to the power of 0, is 1.
A logarithm is another notation symbol we can use to express big numbers. It’s not only for this, but we can look at it as a reverse for exponents. If I have 85, which is 32768, in logarithmic form, we write log subscript the base number, then result, which equals the exponent. So 85 as a logarithm would be log8(32768) = 5. If you want to know the result of a number raised to a power, you’d use exponents. If you have the result and the base number but want to know the power it’s raised to, you use logarithms.
Here’s the formula using variables: logb(a) = c ~ bc = a
a is the argument (the result of the exponent), b is the base number, and c is the exponent
If you don’t put anything as b, it will default to 10. This is known as a common logarithm
We know that log2(8) = 3, but what if I wanted log2(1/8)? It would be -3. When you add dividend 1 over the argument, the positive exponent turns negative.
This really gets messy when the argument is lower than the base, especially when it’s a fraction. If we got log8(1/3), the result would still be in logarithmic form.
Properties
There are 4 properties of logarithms: product, quotient, power, and change of base rule.
The product rule [ logb(MN) = logb(M) + logb(N) ] is when we have multiplication as the argument. Say we have log2(4 * 8), which is 32. We first expand the factors into separate logarithms and add them, log2(4) + log2(8). The first’s exponent is 2, and the second’s is 3, which equals 5. So once we multiply the argument and solve the logarithm, we see that 25 = 32. The same applies that if you start with 2 separate logarithms, you’d have to condense them. And keep in mind this only works if the bases are the same. We’ll get to different bases later.
The quotient rule [ logb(N/M) = logb(M) − logb(N) ] is the inverse of the product rule. If the argument is a division problem, you separate them and subtract.
The power rule [ logb(Mp) = plogb(M) ] is when the argument has an exponent and you switch it to in front of the log to multiply it, which gives the same result. log4(42) = 2log4(4)
Lastly, the change of base rule is used when the argument doesn’t fit the base as a whole number. If I have log2(50), no regular exponent of 2 goes into 50, so the resulting exponent would be messy. This isn’t always the answer, but usually it’s best to change the base to 10 so it’s easier to work with. Then we split the logarithms into a division problem with our new base 10. The argument stays the same for the numerator, and the original base becomes the argument for the denominator. the result would be a decimal, about 5.64.
Grant Abraham 