What are logarithms?

fotaczkak4

fotaczkak4

Answered question

2022-08-08

What are logarithms?
I have heard of logarithms, and done very little research at all. From that little bit of research I found out its in algebra 2. Sadly to say, I'm going into 9th grade, but yet I'm learning [calculus!?] and I don't know what a logarithm is! I find it in many places now. I deem it important to know what a logarithm is even though I'm jumping the gun in a sense. My understanding of concepts, is just like that of programming. In the mean time, you know its there, and your ITCHING SO HARD to find out what that is, but nope! For now we use it, tomorrow we learn what it does.
I just know that to identify a logarithm at my level, I just look for a log. :P

Answer & Explanation

Madilyn Dunn

Madilyn Dunn

Beginner2022-08-09Added 16 answers

If you know what a power function is:
a b = c
you can choose to solve for a or b. If you want a, take b-th root on both sides:
a = c b
Imagine b = 2
However, if you want to get b, you take the logarithm:
b = log a c
Here, I used the logarithm with a base a. Logarithms of different bases are related: they are simple multiples of each other. Common logarithms are log 10 (the base is usually skipped), and the natural logarithm ( log e x = ln x) which is a very nicely-behaved function when you go further in calculus.
The numerical meaning of logarithm can be roughly understood as this: the whole part of the value of log 10 x counts the number of digits in x. For instance
log 1 = 0
log 10 = 1
log 100 = 2
log 1000 = 3
and so on. Of course, you can evaluate something like
log 500 = 2.69...
log 0.05 = 1.30...
The rest of the properties follow from the definition that it inverts a b = c. For instance, logarithm of a product can be split into sum of logarithms:
log a b = log a + log b
Ultimately, it's just another elementary function, like roots, polynomials and so on.
Sandra Terrell

Sandra Terrell

Beginner2022-08-10Added 2 answers

Logarithms are the inverse to exponentiation. Simply put, log a ( k ) is the solution to the equation
a x = k
Here a is called the "base" of the logarithm.
Obviously, k > 0 and therefore, the logarithms of negative numbers are not defined, as long as we are dealing with the set of real numbers. Also 1 x = k will have no solutions for k 1, therefore a must be not equal to 1
( 1 ) x (or any other negative base) isn't well defined for non-integral x and therefore a > 0
This sets some constraints on the domain of log
a 1
k > 0
a > 0
log e ( x ) is called the natural logarithm, sometimes abbreviated as ln ( x ) or simply log ( x )

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