To solve for the function $d(x)$, given that $h(x)$ is the log-log plot (with logarithm base 2) of $d(x)$, we need to use the relationship between a log-log plot and a power-law function.

1. Relationship between log-log plot and power-law function:

• If $h(x)$ is a log-log plot of a function $d(x)$, then: $h(x) = \log_2(d(x))$ This implies: $d(x) = 2^{h(x)}$ You are given that $h(x) = 2x + 6$, so we will substitute this into the formula.

2. Find $d(x)$:

Substitute $h(x) = 2x + 6$ into the equation $d(x) = 2^{h(x)}$: $d(x) = 2^{2x + 6}$ Now, let's simplify the expression for $d(x)$: $d(x) = 2^{2x} \cdot 2^6$ $d(x) = 2^{2x} \cdot 64$ Thus, the function $d(x)$ is: $d(x) = 64 \cdot 2^{2x}$

3. Conclusion:

The function $d(x)$ is: $d(x) = 64 \cdot 2^{2x}$

4. Verifying the result:

To check that this satisfies the original log-log plot:

• For $x = 2$, $h(2) = 2(2) + 6 = 10$, and $d(2) = 64 \cdot 2^{2(2)} = 64 \cdot 2^4 = 64 \cdot 16 = 1024$. So, $\log_2(1024) = 10$, which confirms $h(2) = 10$.

• For $x = 6$, $h(6) = 2(6) + 6 = 18$, and $d(6) = 64 \cdot 2^{2(6)} = 64 \cdot 2^{12} = 64 \cdot 4096 = 262144$. So, $\log_2(262144) = 18$, which confirms $h(6) = 18$.

Do you need any more clarifications or details on the solution?

Other related questions:

1. How do you determine the base of a log-log plot?
2. How do logarithmic transformations affect power-law relationships?
3. Can you explain how the properties of logarithms help in converting between functions like this?
4. How would the equation change if the base of the logarithm was different (e.g., natural log)?
5. What other functions can be analyzed using log-log plots?

Tip: Remember that when working with logarithms in a log-log plot, any function $d(x) = b \cdot x^{-a}$ will transform to a straight line with slope $-a$ and y-intercept $\log_b$ of $b$.