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
It undoes exponentiation. If we consider the function f_a(x) = a^(x) where a is some positive real we have an injective function. We can see its by writing:
a^(x) = a^(y)
a^(x) a^(-y) = 1
a^(x-y) = 1
and we know that a^0 = 1 (except when a = 1, in that case a^x = 1 for every x) so:
x-y = 0
x=y
So f_a(x) is injective and so has an inverse as long as a isn’t 1.
Lets call the inverse f^(-1)_a.
So f^(-1)_a(f_a(x)) = x. Great!
If we say have an equation:
a^x = b
we can solve it with the inverse
x = f^(-1)_a(b)
This function is special and frequently used enough to deserve its own name the logarithm. So log_a(y) is the inverse of a^x. The convenient exponential identities work backwards with them, lets write ab in a tricky way:
ab = exp(log(a)) exp(log(b)) = exp(log(a) + log(b))
now take the log of both sides
log(ab) = log(a) + log(b)
So lets adress what base of log and exp we should use. The answer that it doesn’t matter, there is a convenient pick e, but why and what value it has doesn’t matter right now. So lets select e so
exp(x) = e^x and ln(x) = log_e(x)
The statement is that you can get any base with these but thats true for any number so for now e is just a placeholder for your favourite number.
We can see how:
a^(x) = exp(x ln(a)) = exp(ln(a))^(x) = a^(x) = exp(ln(a^(x))) => ln(a^(x)) = x ln(a)
and with that:
x ln(a) / ln(a) = ln(a^(x)) / ln(a)
x = ln(a^(x)) / ln(a) => ln(f_a(x))/ln(a) = log_a(y)
So we now see how you can use just ln to get any kind of logarithm. And now we can adress the issue with f_1(x) now having an inverse:
log_1(y) = ln(y)/ln(1)
and
1 = e^0 => ln(1) = 0
so
log_1(y) ~ 1/0 and so is undefined.
The complex logarithm is slightly more exciting and you’d basically get there with the exponential similarly to how we started operating with it in the last part. Hopefully seeing how you can work things out from just the concept of what this thing is helped making the little fella less mysterious.
It is roughly counting the number of 0’s (or digits) in a big number. 1,000,000 is 6 zero’s 1,000 is 3 zero’s. To the question, how many basketballs can you fit in a school bus? You’re looking to get the number of digits correct rather than being within 5% of the actual answer.
In addition to the good points made about exponents, logarithms have the extremely useful property of transforming multiplication and division into addition and subtraction. This is of keen interest to someone who has to do all their calculations by hand, since it’s easier to add than it is to multiply, and much easier to subtract than divide.
I thought that I understood logarithms, but after reading some of these responses I’m no longer sure.
Logarithm is the inverse of exponentiation, for example:
10^2 = 100, log(100) = 2.
Note, for the logarithm you have to specify the base, for example:
2^5 = 32, log2(32) = 5, but log10(32) ~1.505…
Logarithms are sometimes used for plotting data that grows exponentially. For example if you were to plot inflation vs time, you’d find the growth so rapid you couldnt adequately visualize data from 50 to 100 years ago. Plotting the logarithm of inflation vs time, however, clearly shows a long term trend.
The mathematics goes way beyond these examples, so would only use them as a starting point. For example, you can describe rotations using logarithms, but the math can be advanced.
“X = loge y” was originally e^x = y
Look at the second equation for a sec. First, think about y. As x increases linearly, y increases faster and faster. That’s exponential. Now look at x, as y increases linearly, x increases, but it does so slower and slower. That’s logarithmic. The first equation is how the curvature of that trend is denoted. “Loge”
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.
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.
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.
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