An investment loses 2.3% of its value every week. find the half-life in this situation.

The half-life would be 29.79 weeks.

The amount of an investment is given by 



where A is the total amount, p is the amount invested, r is the interest rate as a decimal number, and t is the amount of time.  In our case, we do not know the principal; r is -2.3% = -2.3/100 = -0.023; and we do not know t.  We do know that we are trying to find when A, the amount of the investment, is equal to 1/2 of the principal; that means A = 0.5p:



We can divide both sides by p, which will cancel it on both sides:


To cancel the exponent t, we will use logarithms: