An element is decaying at a rate of 12% per hour. Initially we have 100g. When will there be 40g left?

Clearly, the first two answers give you the correct answer for how long it takes to reach 40 grams of your element.  But I think it might be helpful to explain a little bit about why you should set your equations up the way they did.

If you think about it, what you are doing here is just like figuring the interest on an investment (only you are getting negative interest since your "investment" is decaying, not growing).  When you try to figure that, you use the formula

A = P(1+r)^n

where A is the final value of your investment, P is the original principal, r is the rate of interest, and n is the number of times the interest is compounded.

In your problem, you start with 100g -- that's your original investment.  It decays at 12% -- that's your interest rate (-.12).  You want to end up with 40g -- that's your final value.  You are asked to solve for n -- how many interest periods (in your case, hours).

So you start with

40 = 100 (1-.12)^n

From there, you do what the first two answers did and you get your result.

I hope this explanation is helpful.