Question

Let f(x) = cx^k be a power function such that f(13) is four times the size of f(1). What is the power k?

Answer 1

Answer:

0.540

Step-by-step explanation:

Hi there,

To get started in order to solve this, please recall the property of logarithms.

This question is a bit tricky, but doable. Best way to solve this is first compare the two functions:

$f(x) = cx^{k} \\ f(13)=4f(1)$  

Now, let's see what the function gives at 13 and 1:

$f(13)=c13^{k}\\f(1)=c1^{k}$

Something to recognize is that 1 to the power of anything is just 1, so f(1) is reduced to:

$f(1)=c$  We are making progress!

Next, we can set both f(13) forms equivalent to each other, watch this:

$f(13)=f(13) = 4f(1)$ but we know what f(1) is equal to, and let's substitute our knowns:

$c13^{k}=4c1^{k} = 4c$ the c constant on both left and right side cancel out:

$13^{k}=4$ Now, take the logarithm form of both side of the equation. I used natural log, but you can also use common logs:

$ln(13^{k})=ln(4)$  ⇒ $k*ln(13)=ln(4)$ this was performed using the power  log rule.

Isolate k:

$k=\frac{ln(4)}{ln(13)} =0.540$ using a calculator.

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thanks,