Radioactive substances decay over time. The mass $M$, in grams, of a particular radioactive substance $d$ days after the beginning of an experiment is shown in the table below.

$$\begin{array}{|c|c|} \hline \text{Number of days, } d & \text{Mass}, M \text{ (grams)} \\ \hline 0 & 45.00 \\ \hline 30 & 40.57 \\ \hline 60 & 36.58 \\ \hline 90 & 32.98 \\ \hline \end{array}$$

If this relationship is modeled by the function $M(d)=a\cdot 10^{bd}$, which of the following could be the values of $a$ and $b$?

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We can first use the row $d=0, M=45$ to find the value of $a$:

$$M(0)=45$$ $$M(d)=a\cdot 10^{bd}$$ $$45 = a \cdot 10^{b(0)}$$ $$45 = a \cdot 10^0$$ $$45 = a$$

Substitute this value back into the initial equation:

$$M(d)=45\cdot 10^{bd}$$

We can use another point, such as $d=30, M=40.57$ to find the value of $b$:

$$M(30)=40.57$$ $$40.57=45\cdot 10^{b(30)}$$ $$40.57 = 45 \cdot 10^{30b}$$ $$\frac{40.57}{45}=10^{30b}$$ $$0.90=10^{30b}$$

We can test options 3 and 4 to confirm which one is the correct choice, or we can use logarithms.

Solving with logarithms:

$$0.90=10^{30b}$$ $$\log{0.90}=\log{10^{30b}}$$ $$\log{0.90}=30b$$ $$b = \frac{\log{0.90}}{30}$$ $$b \approx -0.0015$$