Exponent difference

Exponent difference is the mathematical operation in which two exponents are given and base number and power number are switched then finding the difference between the resulting numbers. The operation sign for this operation is ~. For example, the given numbers are 2 and 5 (2~5) and forms the exponent 2$5$ and the numbers that make up power and base swap to 5$2$. 2$5$ is equal to 32 and 5$2$ is equal to 25 and subtracting these two resulting numbers would yield 7, which is the same as the sum of 2 and 5. To make sure that the operation is not the same as the sum, let's do 3~4. 3$4$ is 81 and 4$3$ is 64 and the difference is 17. The exponent difference gets greater when the difference between two numbers gets greater.

The exponent difference involving the zero in a given number would always be 1 or −1 since anything to the power of zero is 1 and zero to the power of any number is always zero. The exponent difference involving the one in the given number would always result in the exponent difference being the same as the regular difference between the two numbers, e.g. 3~1 is 2 since 3$1$=3 and 1$3$=1. The exponent difference using two identical numbers would always result in zero since switching numbers in base and power would have the same exponent, therefore subtracting two identical numbers is zero.

The exponent difference can be negative, e.g. 5~2 is −7, if the number to the left of tilde is greater than the number on the right, with the exception for the 4~2, 3~2, and the problems with the right number being 1 or 0. The operation using 4 and 2 as the given numbers is the only non-identical pair in which the result is zero, as 4$2$ and 2$4$ both equal same number (16). The operation using 3 and 2 as the given numbers is the only operation with none of the given numbers being 0 nor 1 in which the result is the same as the difference between the two given numbers as 3$2$ is 9 and 2$3$ is 8.