1. **State the problem:** We want to find the value of $b$ in the exponential function $y = ab^x$ given points $(0,1.5)$ and $(1,3)$. 2. **Recall the formula:** The function is $y = ab^x$ where $a$ is the initial value and $b$ is the base. 3. **Find $a$ using the point $(0,1.5)$:** Since $x=0$, $y = a b^0 = a \times 1 = a$, so $a = 1.5$. 4. **Use the point $(1,3)$ to find $b$:** Plug in $x=1$, $y=3$: $$3 = 1.5 \times b^1 = 1.5b$$ Divide both sides by 1.5: $$\frac{3}{\cancel{1.5}} = \cancel{1.5}b \Rightarrow 2 = b$$ 5. **Interpretation:** The value $b=2$ means the function doubles each time $x$ increases by 1. 6. **Final answer:** $b = 2$