Question

Suppose that a regression line for some data transformed with logarithmspredicts that when y equals 8, log(%) will equal 1.603. What does theregression line predict y will equal when y equals 8? Round your answer to thenearest whole number.

Answer 1

Given the relationship between y and x to be

$y=a^x\text{ ------ equation 1}$

Take the logarithm of both sides,

$\begin{gathered} \log y=\log ^{}_{}a^x \\ \Rightarrow\log \text{ y = x }\times\text{ log a ---- equation 2} \end{gathered}$

But when x = 8, log y = 1.603.

Thus, substituting the above values into equation 2, we have

$\begin{gathered} 1.603\text{ = 8 }\times\text{ log a} \\ \text{divide both sides by 8} \\ \log \text{ a= }\frac{1.603}{8} \\ \Rightarrow\log \text{ a =0.2}004 \\ \text{Thus, } \\ a=1.586 \end{gathered}$

From equation 1,

$\begin{gathered} y=a^x \\ \Rightarrow y=1.586^x\text{ ----- equation 3} \end{gathered}$

Thus, when x = 8

$\begin{gathered} y=1.586^x \\ y=1.586^8 \\ \Rightarrow y=40.03 \end{gathered}$

Thus, the value of y will be 40 (to the nearest whole number)

The correct option is D