So, I am going into college as a math major with a concentration in secondary education, but despite how much I’ve learnt and understood, the one thing I’ve never been able to wrap my head around is logarithms and how they work, what they do, and things such as that. Anyone know a good way to explain it?
Extraneous questions:
what is *ln*?
What does a subscript after the log mean, especially if it’s not 10?
In: 0
Logarithms are the opposite of exponents.
Just like the opposite of multiplication is division, or the opposite of addition is subtraction, the opposite of putting something to a power is the logarithm.
If a = b + c you can reverse it with b = a – c
If a = b × c you can reverse it with b = a ÷ c
If a = c^b you can reverse it with b = log_c(a) (log base c of a)
`Ln` is shorthand for log base `e`. It exists because there’s so much maths around y = e^x etc.
Logarithms are the opposite of exponentiation (raising to a power)
Think of a number triangle like this:
3
2 8
so you read that as 2 to the power 3 == 8 (2^3 = 8)
for logarithms, it’s the other way around. log2 8 == 3
so that means what power of 2 will give you 8.
The subscript is the ‘base’ number.
also third leg of triangle: 3rd (cube) root of 8 == 2
ln is a special version of log (called the natural log) where the number e is the base power.
Almost every math function has an inverse, or a second function that can undo the first one. Addition and subtraction, multiplication and division, exponents of a variable and roots. This is how we can solve for an unknown variable when we know the result in an equation. But what about exponents *with* a variable (10^x instead of x^10)? That’s what a logarithm is for, to be the inverse of exponents with a variable.
We know 100 is 10^2, so log(10) of 100 is 2. Similarly, log(10) of 1000 is 3. If something grows at 10 times itself each day, the logarithm will describe how many days it would take to get there from 1. But chances are that something real doesn’t suddenly multiply at the end of the day. If we want to know when and during which day it reaches 200, we can put 200 into log(10) and we [get about 2.3](https://www.wolframalpha.com/input?i=log10+200). It would take 2 days and about 7.2 hours to go from 1 to 200 in this situation.
log(10) is different than just log because different logarithms are for different bases. log(10) is for 10^x, log(2) is for 2^x, and the specially named ln (stands for natural logarithm) is for *e*^x.
Logarithms are the inverse of exponents, similar to how division is the inverse of multiplication.
If you have an exponent of the form x^y = z
Then the corresponding logarithm is
log(x) z=y
Basically you are asking how many times to I have to multiple x by itself to get z.
x is called the base of the logarithm. If it’s not specified it’s usually assumed to be base 10. The exception is the natural logarithm, usually abbreviated ‘ln’ instead of ‘log’. The natural logarithm use the mathematical constant e as it’s base.
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