What is a logarithm?
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A base raised to an exponent gives you a number. But suppose you only have the base and the result. You would use a logarithm to figure out what the exponent had been. That’s where it comes from. In terms of why it’s useful, it’s not so much about solving for the exponent of particular equations as much as it is about modeling relationships between numbers that experience exponential growth or decay. So you would use log(x) a lot in describing those things, kind of like how you would use doubling a lot, or 2(x), to describe a process like mitosis.
Let’s say you have a farm with a pen holding three rabbits.
Let’s say every month, the population of rabbits grows by three.
How many new rabbits do you have after five months? You can use **multiplication** to find this answer: 3 × 5 = 15 new rabbits.
How many months does it take to get fifteen new rabbits? You can use **division** to find this answer: 15 ÷ 3 = 5 months.
Now let’s say instead that every month, the population of rabbits triples.
How many rabbits do you have at the start of the fifth month? You can use **exponents** to find this answer: 3^5 = 243 rabbits.
At which month will there be 243 rabbits? You can use **logarithms** to find this answer: log₃(243) = 5th month.
It’s the opposite of an exponent. A root is just a fractional exponent, not the opposite. Sqrt(x) = x^1/2])
If I had a^x = b, I can’t do the xth root of both sides, so I do a log.
If I do the log base a (loga) then I can get x = loga(b)
Log2(8) is asking 2 to what power is 8, the answer is 3 because 2^3 is 8.
Ln(e^2 ) is asking e to what power is e^2 , ans the answer is 2.
How many digits does it take to write out the number? That’s the logarithm.
Technically only log base 10, and doesn’t deal with partial digits, but that’s the intuitive answer.
If you have an exponential function like f(x) = 10^x, that function turns an additive increase in input, into a percentage increase in output.
So if your input goes from 0 to 1, f(x) increases from 1 to 10. If your input goes from 1 to 2, f(x) increases from 10 to 100. If your input goes from 2 to 3, f(x) increases from 100 to 1000. And so on.
A logarithm function is the reverse (technically, the inverse) of an exponential function. So if g(x) = log(x), then if you increase your *input* from 1 to 10, the *output* of g() increases from 0 to 1. If you increase your input from 10 to 100, the output of g() increases from 1 to 2. If you increase your input from 100 to 1000, the output of g() increases from 2 to 3.
To explain, I’m going to start with subtraction and division.
Subtraction is the “reverse” of addition. Addition is figuring out 3 + 5 + ___. Subtraction is figuring out 3 + ___ = 8, or __ + 5 = 8. By moving which number is left blank, you switch the question from an addition problem to a subtraction problem. If you want to rewrite them as proper subtraction problems, you get 8 – 3 = ___ and 8 – 5 = ____
In the same way, division is the “reverse” of multiplication. A multiplication problem is 3 * 5 = ___; and two division problems are 3 * ____ = 15 and ____ * 5 = 15. Again, moving which spot is left blank. To rewrite them: 15 / 3 = ____ and 15 / 5 = _____
The next “level” of arithmetic is powers or exponents. And exponential problem is 3^(5) = ____ (Sometimes written 3 ^ 5 = ___). A key thing to note here is that, unlike the previous two “reverse”s, in this case, you can’t switch the numbers: 3^5 isn’t the same as 5^3. For this reason, you need two different “reverse” functions.
Logarithms are one of the two “reverse”s. In this case, 3 ^ ___ = 243. Rewritten, this is Log3(243) = ____
The other “reverse” is roots. In this case, ___ ^ 5 = 243. Rewritten, Root5(243) = ___.
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To go back to your question, a logarithm is asking “Base raised to what power gives this number?”
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