What is the result of exponentiation?

[krausr79]

If you add, subtract, multiply, or divide things you will end up with a sum, difference, product, or quotient.

What is the result of exponentiation?
What is the result of a logarithm?

 

[LCKurtz]

There is a conceptual difference here. Notice that the sum, difference, product, and quotient all require two arguments. They are examples of binary operations.

Your examples of logarithm and exponentiation are unary operations, requiring only one argument. The name of the operations is usually used to describe the result:

"the logarithm of k" or "log [to the base b] of k" or "natural log k".
"exponential k" or "the exponential of k [to the base b]" or "b to the k^th" power.

 

[Borek]

Your examples of logarithm and exponentiation are unary operations, requiring only one argument.

At the risk of hijacking the thread - this statement make me wonder...

In the usual notation - log(x) - we write logarithm as unary operation. But this notation is ambiguous and can refer to any logarithm, of any base. It stops to be ambiguous when we explicitly mention base - log₂(x). But this in fact means we have two arguments, something like log(base,x). So to be precise we probably should write "logarithm to base b is unary".

 

[LCKurtz]

At the risk of hijacking the thread - this statement make me wonder...

In the usual notation - log(x) - we write logarithm as unary operation. But this notation is ambiguous and can refer to any logarithm, of any base. It stops to be ambiguous when we explicitly mention base - log₂(x). But this in fact means we have two arguments, something like log(base,x). So to be precise we probably should write "logarithm to base b is unary".

Technically probably so. In most situations the context implies what base log is understood to use. One could argue that without explicit or implicit understanding of the base, the function isn't defined. Only after the base is specified have we defined a function, and it is unary. At any rate, I don't think that is what was bothering the OP.

 

[krausr79]

I disagree. These are not unary operations. Logarithms require both a base and an argument, and for exponentiation, its a base to an exponent.

Log₁₀(1000)=3
10³=1000

I suppose you could make a case that a logarithm is unary once the base is nailed down (a la calculators), but exponents??

 

[joeoettinger]

Here's a quote from a Utah math dept webpage:

+ addition. The result of an addition is a sum .
- subtraction . The result of a subtraction is a difference .
* multiplication . The result of a multiplication is a product .
/ division . The result of a division is a quotient or a ratio . The number that is being divided is the dividend, the number it is being divided by is the divisor.
^ or ** exponentiation . The result of exponentiation is a power .

http://www.math.utah.edu/online/1010/precedence/

But I think that's wrong. I think power is usually another name for exponent. "a to the power b" is the same as "a to the exponent b".

Though it's unsatisfactory, maybe it's best to call the result and process by the same term: "The exponentiation of a to the b is y".

In English, the result of a process is often given the name of the process, for example, "He was so hungry he ate two servings."

Expressing math in English is awfully hard. Hence mathematical notation.

 

[logics]

. The result of exponentiation is a power .
. I think power is usually another name for exponent.
In English, the result of a process is often given the name of the process.

According to SOED [p.2371], it is the other way round. Utah page is right.
"power" ([III12 Math] the primary meaning is): "A value obtained..." namely the result, then "...Also, an exponent."
second power [is explained as] (the square of a quantity)

 

[logics]

What is the result of exponentiation?
What is the result of a logarithm?

... this in fact means we have two arguments,...

that is right: a logarithm is the result of the reverse operation of exponentiation : is the exponent, (as the factor[/quotient] is the result of the reverse operation of multiplication 2*3→6, 6:2 → 3)
exponentiation: 2^[log=]3 → 8, reverse operation : log₂ 8 → 3.