The image shows a problem related to the exponential function in a semi-logarithmic plot. The function is of the form $h(x) = ab^x$, where the graph passes through the coordinates $(0, 0.4)$ and $(4, 5)$. The task is to determine the values of $a$ and $b$ in the exponential function, given that the x-axis has linear scaling and the y-axis has a logarithmic scale with base 10.

Let's go step-by-step to solve for $a$ and $b$:

1. General form of the exponential function: The equation is given as: $h(x) = ab^x$

2. Substitute the given points into the equation:

   • For the point $(0, 0.4)$, substitute $x = 0$ and $h(0) = 0.4$: $0.4 = ab^0 \quad \text{(since anything raised to the power of 0 is 1)}$ $0.4 = a$ So, $a = 0.4$.

3. Substitute the second point $(4, 5)$:

   • For the point $(4, 5)$, substitute $x = 4$ and $h(4) = 5$ into the equation: $5 = 0.4b^4$ Solve for $b$: $b^4 = \frac{5}{0.4} = 12.5$ $b = \sqrt[4]{12.5} \approx 1.861$

Thus, the values of $a$ and $b$ are:

• $a = 0.4$
• $b \approx 1.861$

Let me know if you'd like further clarification or help with other related problems!

Related questions to explore:

1. How do logarithmic plots help in interpreting exponential functions?
2. What are the different forms of exponential functions used in mathematical modeling?
3. How does changing the base $b$ in an exponential function affect its growth rate?
4. Can exponential functions be applied in real-world scenarios, like population growth or finance?
5. How do you solve for parameters in exponential equations with other points on the curve?

Tip: When working with exponential equations, remember that logarithmic transformations can help linearize the data, making it easier to find the parameters.