To find the exponential equation that best fits the given data in the form $y = ab^x$, we'll use logarithmic transformation and linear regression. This involves taking the natural logarithm of $y$ values and then performing a linear regression on $x$ and $\ln(y)$.

Step-by-step Process:

1. Transform the exponential model:

   • Given: $y = ab^x$
   • Taking the natural logarithm on both sides: $\ln(y) = \ln(a) + x \ln(b)$
   • This equation resembles a linear form: $\ln(y) = m \cdot x + c$, where:
     • $m = \ln(b)$ (slope)
     • $c = \ln(a)$ (y-intercept)

2. Calculate $\ln(y)$ for each $y$ value:

   • $x = [1, 2, 3, 4, 5, 6]$
   • $y = [1044, 1492, 2408, 3516, 4742, 6853]$

3. Perform linear regression on $x$ and $\ln(y)$.

I'll proceed with the calculation to find the values of $a$ and $b$.The exponential equation that best fits the given data is:

$y = 729.43 \cdot 1.46^x$

where:

• $a = 729.43$ (rounded to two decimal places)
• $b = 1.46$ (rounded to two decimal places)

Would you like a detailed explanation of the steps, or do you have any other questions?

Here are some related questions:

1. What is the correlation coefficient ($r$) of the regression?
2. How do we determine if an exponential model is the best fit for the data?
3. How does transforming data help in regression analysis?
4. Can you solve for $y$ when $x = 7$ using this model?
5. How would the process differ for a logarithmic or linear model?

Tip: Exponential models are suitable when the rate of change of a variable increases or decreases proportionally to its current value.