what is the logic/reason behind the mathematical fact that 10^0 equals 1?

Answer 1

Many definitions in mathematics are made so that there will be consistency among different properties/rules.

10^0 arises from the definition x^0 = 1 (provided x does not equal zero). Either definition arises from the following


We know that any number divided by itself = 1. That is 10/10 = 5/5 =x/x (provided x dos not equal zero) and so on.

We also know using the properties of exponents that when we divide using exponents we subtract the denominators's exponent from the numerator's exponent. That is x^5/x^2 = x^(5-2) = x^3. This holds true so long as the base used in the numerator and denominator are the same, example 10^5/10^2 = 10^(5-2) = 10 ^3

How then should 10^0 or x^0 be defined so that aforementioned two ideas coexist?

10^n/10^n = 10^(n-n) = 10^0 however

10^n/10^n is also a value divided by itself which we know to be 1.

Thus 10^0 = 1 (or by the same argument x^0 = 1 provided x does not equal zero).

Answer 2

If you just plotted the curve y = 10^x, you'll see that where x is a large positive number, y is very large, and where x is a large NEGATIVE number, y approaches 0. This nice, well behaved curve intersects the y-axis where x = 0 at where y = 1. It would be very strange if it didn't intersect the y axis at y = 1.

Answer 3

10^a/10^a = 1, and the left side = 10^(a - a) = 10^0, and so
10^0 = 1.

In general, x^a/x^a = x^(a-a)= x^0 = 1.

Answer 4

Set 10^0 to an unknown and solve for it:

10^0 = x

Take the log of both sides:

log(10^0) = log(x)

0 * log(10) = log(x)

Zero times anything is zero:

log(x) = 0

x = 1

Answer 5

Log 1 (to the base 10) = 0
So, 10^0 = 1
It is based on the laws of logarithms.

Answer 6

n^x/n^x = n^(x - x) = n^0

as you notice whenever you have a number divided by itself, you will get 1, so thats why 10^0 = 1

Answer 7

think in terms of a series -

10^3=1000
10^2=100
10^1=10
10^0 =.....