Question: inverse exponential processing, say N = 10000, N=10^x. How can we solve for value of X?
For this simple case, write N as a power of 10.
viz. N = 10,000 = 10^4.
i.e. N = 10^4 = 10^X
Hence X = 4.
For a non-simple case, where N is not an integer power of 10, then use logs.
We would have,
N = 10^X
ln(N) = ln(10^X) = X*ln(10)
Hence, X = ln(N)/ln(10)
e.g. N = 15,321, then
ln(15321) = X*ln(10)
X = ln(15321)/ln(10) = 9.6369797/2.302585
X = 4.185287
If you put that value in a calculator, and then raise 10 to that power, you will get 15,321 in your results display.