Question

A baseball player has been improving every season since making it to the big leagues. Below is a table of the runs he has scored. His manager wants to try to determine when he will score 243 runs. Explain how to create the exponential function that represents his run-scoring abilities. Then explain how to convert this function into a logarithmic function and why this can help the manager answer his question.

Answer 1

@jim_thompson5910 can you help?

Answer 2

@Concentrationalizing can you?

Answer 3

Well, a general exponential function would look like \(a^{x}\). Given that, let's see if we can try and determined what our a-value would be based on the chart you have. The season will represent x-values and the runs will represent y-values. So if we plug those values in:
\(a^{1} = 3\)
\(a^{2} = 9\)
\(a^{3} = 27\)

So what would a be?

Answer 4

a would be 3, right?

Answer 5

Right. So we can say our function is \(y = a^{x}\). Now do you know how to change an exponential equation into a logarithmic one?

Answer 6

No I don't

Answer 7

Okay. Let's use this idea then.
\(log_{b}m = x \implies\ b^{x} = m\)
Maybe you've seen that connection between exponentials and logarithms before.

Answer 8

Either way, you can kind of see how the variables in that identity switch around when you want to change from a logarithm to an exponential and vice-versa

Answer 9

So in the logarithm would b be 3?

Answer 10

Correct

Answer 11

And then you could say y would take the place of m, so that would give you

\(log_{3}y = x\)

Answer 12

So the identity above would be your explanation as to why you can claim that this logarithmic function works. As for funding your answer, that comes down to solving for x in the equation \(3^{x} = 243\)

Answer 13

x would be 5\[3^{5}=243\]

Answer 14

Yep, so 5 seasons to score 243 runs.

Answer 15

Okay, I think I understand now. Thank you :)

Answer 16

No problem :)