I mean, I’ve used them in algebra many times but I never really understood what it does. Kinda like in biostatistics how I could do the math, but how it worked was beyond me entirely.
So yeah: like what’s this sorcery and what does it do/why do we use it?
In: Mathematics
A log asks how many times you need to multiply 1 number to get to another number.
For example, log2(8)=3 because you need three 2s (2×2×2) to get 8.
A witch gives you a bag of n marbles. One of them is lucky. If you touch it, you will shit gold.
So you divide the marbles into m equal-ish piles, differing by no more than 1 in size, and give each to a friend.
Then, each of your friends divides their pile into m equal-ish piles and gives each of these to a friend.
This continues until nobody has more than one marble.
logm(n) = the # of 5yos who can now pay off their parents’ student loans (rounded up or down, as luck would have it)
My favorite way for conceptualizing a log scale is to imagine a brick wall, where the position of a brick from left to right corresponds to some value you’re plotting.
If you look at the brick wall straight on, it looks like [this](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRn-wd2YsRsF-sTytPPbOzLnjpIuY627CS-6Q&s). This is like plotting values on a linear scale: the relative width of a brick in the picture is the same everywhere. This is good for comparing values that are of similar size, but if our dataset contains both tiny and huge numbers, like 10, 20 and 34512, we’d need to extend the brick wall hugely from left to right in order to show all values at the same time. From such a zoomed-out view, 10 and 20 would be so close together that you couldn’t tell them apart, even though 10 is twice as big as 20.
If you look at the brick wall at an angle, it looks like [this](https://www.shutterstock.com/image-photo/brick-wall-angled-view-260nw-462391810.jpg). This is like plotting on a log scale. Now, the further to the right in the image you look, the more bricks fit into the same space on screen. This way, you can keep both bricks that are close to you (low values) and bricks that are far from you (high values) in your view at the same time.
I have no idea if this makes sense to anyone other than me.
It’s the operation that transforms multiplications into additions : log(a * b) = log(a) + log(b).
Two classic applications come to my mind:
**Logarithmic scale:** It can be useful if you’re working with something that grows “geometrically”, that is, through successive multiplications, so gets big very quick, and doesn’t fit on your graph paper sheet. Put it through the logarithm, and lo!, now it grows “arithmetically”, that is through successive additions, that’s more manageable to plot. Examples: Decibel scale for sound levels, infectious disease transmission formula.
**Inverse of power functions:** You know that 2 to the power of 5 is 32. Now if you know that 32 is a power of 2 but want to know which one, logarithm gives you the answer.
Here’s how I think about it: loga(b) is the power that makes ‘a’ become ‘b’. Which implies: a^(loga(b)) = b, we don’t even need to know the value of loga(b) in the previous equation, just that ‘a’ multiplies by itself loga(b) times to become ‘b’.
A good way to think about what a logarithm DOES, is they tell you how “long” a number is. If log(100) is 2, and 1000 is 1 digit longer, log(1000) is 3. The function just also can tell you numbers in between as a smooth function.
Where was all of this explanation in high school?
In addition to what others have said, it can be thought of as repeated division.
To calculate log base 2 of sixteen, divide by two repeatedly, until you get to one. It take 4 times, so log base 2 of 16 is 4.
I hate that some people just treat it like some sort of magic trick that undoes exponents.
In addition to all the other mentions: inverse of exponents, plot scaling, statistical data normalization, summing log values to multiply. One thing that I rarely see mentioned, that makes them far more useful is that you can easily get the log for any base number, by dividing the log of the number by the log of the base you want. instead of knowing what you need to put 10 to the power of, you can know what any number needs as an exponent to get the number in question. If you want to know how long it takes to double your money with 10% annual interest. log2/log1.1 will tell you, 7.27 years.
You know how exponents work, right? Some number taken to a power (call it “n”)is the same as multiplying the number by itself n times.
A^2 = A x A
A^3 = A x A x A
A^4 = A x A x A x A
But what if the exponent (2, 3, 4, etc.) could be a non-integer like 2.6 or 3.524? That exponent is the logarithm.
You can express any number like this.
10^2.65 = 446.68
In this example 2.65 is the Base10 logarithm of 446.68
Here’s the real beauty of logarithms: adding logarithms is the same as multiplying the numbers.
10^2.65 = 446.68
10^1.37 = 23.44
What happens if you multiply 446.68 x 23.44? The answer is 10,470
What happens if you add 2.65 + 1.37?
The answer is 4.02
10^4.02 just happens to be 10,470.
In fields like radio frequency (RF) engineering, we use this property all the time. We just multiply the logarithm by 10 and call it “dB” units. This way you can add logarithms all day long, and it’s the same as multiplying the numbers. Adding is much easier than multiplying!
Fun fact: the fret spacing on a guitar fingerboard is logarithmic.
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