Let's analyze the question together.

We are given the equation:

$\log y = 2(a + \log x),$

and asked to find the values of $a$ and $b$. The diagram shows a graph of $\log y$ plotted against $\log x$, with two points labeled:

1. At $(2, 3)$: $\log x = 2$, $\log y = 3$.
2. At $(5, b)$: $\log x = 5$, $\log y = b$.

Step 1: Rearrange the equation into a straight-line form

The given equation can be rewritten as:

$\log y = 2a + 2\log x.$

Comparing this with the equation of a straight line, $y = mx + c$, we see:

• The slope ($m$) of the line is $2$.
• The $y$-intercept ($c$) of the line is $2a$.

Step 2: Use the point $(2, 3)$ to find $a$

Substitute $\log x = 2$ and $\log y = 3$ into the equation:

$3 = 2a + 2(2).$

Simplify:

$3 = 2a + 4.$

Solve for $a$:

$2a = 3 - 4,$

$2a = -1,$

$a = -\frac{1}{2}.$

Step 3: Use the slope to confirm $b$

The slope of the line is $2$, so the change in $\log y$ with respect to $\log x$ is $2$. Between the two points:

• First point: $(2, 3)$,
• Second point: $(5, b)$.

The slope formula is:

$\text{slope} = \frac{\Delta (\log y)}{\Delta (\log x)} = 2.$

Substitute the values:

$2 = \frac{b - 3}{5 - 2}.$

Simplify:

$2 = \frac{b - 3}{3}.$

Solve for $b$:

$b - 3 = 6,$

$b = 9.$

Final Answer:

• $a = -\frac{1}{2}$,
• $b = 9$.

Would you like further clarification or additional details?
Here are 5 related questions to explore further:

1. How do we calculate the slope of a line from two points?
2. Why is the equation rearranged into a straight-line form?
3. What are the key differences between linear and logarithmic scales?
4. How does the value of $a$ affect the graph of $\log y$ against $\log x$?
5. How can you use graphing to verify the calculated values of $a$ and $b$?

Tip: Always verify your results by substituting back into the original equation!