The half-life would be 29.79 weeks.
The amount of an investment is given by
[tex]A=p(1+r)^t[/tex]
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:
[tex]0.5p = p(1-0.023)^t[/tex]
We can divide both sides by p, which will cancel it on both sides:
[tex]0.5 = (1-0.023)^t
\\
\\0.5=0.977^t[/tex]
To cancel the exponent t, we will use logarithms:
[tex]log_{0.977}0.5=t
\\
\\t=29.789\approx 29.79[/tex]
