Hi Marsia,

One way to see the relationship is to use a calculator. To make sure we are using the same calculator try the online calculator at http://www.globalrph.com/calc.htm I wanted a calculator with a log button (this is log base 10) a 10^x = 10^x button a ln button (this is log base e) and an e^x = e^x button. So try these examples

1. Type 5 then press log then press 10^x. Record your answer. (There is roundoff error in this calculator so round up to an integer.)
   
   
2. Type 3 then press log then press 10^x. Record your answer.
   
   
3. Type 5 then press 10^x then press log. Record your answer.
   
   
4. Type 3 then press 10^x then press log. Record your answer.

Try some other numbers and then repeat the examples using ln and e^x.

What you should see is that if you start with a number x, find y = log(x) and then calculate 10^y your answer is x, the number you started with. Also backwards, if you start with a number s, find t = 10^s and then calculate log(t) your answer is s, the number you started with. Using mathematical notation what you have seen is

10^log(x) = x and log(10^x) = x and e^ln(x) = x and ln(e^x) = x

These facts are true regardless of which logarithm function you use, that is

b^log_b(x) = x and log_b(b^x) = x for any positive integer b.

The only possible problem is that when calculating b^log_b(x) = x you must start with a positive x as the domain of the log function is all positive x.

Penny