We have: #0.09 = 12 cdot 0.5^(t) - 1#
First, let's add #1# to both sides of the equation:
#Rightarrow 0.09 + 1 = 12 cdot 0.5^(t) - 1 + 1#
#Rightarrow 1.09 = 12 cdot 0.5^(t)#
Then, let's divide both sides by #12#:
#Rightarrow frac(1.09)(12) = frac(12 cdot 0.5^(t))(12)#
#Rightarrow 0.09083333333 = 0.5^(t)#
Now, let's apply #ln# to both sides:
#Rightarrow ln(0.09083333333) = ln(0.5^(t))#
Using the laws of logarithms:
#Rightarrow ln(0.09083333333) = t ln(0.5)#
Finally, let's divide both sides by #ln(0.5)#:
#Rightarrow frac(ln(0.09083333333))(ln(0.5)) = frac(t ln(0.5))(ln(0.5))#
#Rightarrow frac(ln(0.09083333333))(ln(0.5)) = t#
#therefore t = frac(ln(0.09083333333))(ln(0.5))#
