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Zepdrix
The log function actually grows slower than any power of x. So we can throw option B out the window right away.

Exponential functions grow much much faster than powers of x. So we can throw away option D.

It's hard to compare exponentials which have different bases. We can apply a fancy little trick remembering that the log and exponential are inverse operations of one another.

[tex]\rm y=e^{ln y}[/tex]

We can use this algrebraic technique to rewrite the 3^x in terms of a base e.

[tex]\rm (3^x)=e^{ln(3^x)}[/tex]

Apply one of your exponent rules to bring the x down in front of the log,

[tex]\rm =e^{(ln 3)x}[/tex]

This is easier to compare with e^(4x) now.

Which coefficient is larger? 4 or (ln3)?
It turns out the 4 is larger!

So e^(4x) will grow the fastest.

If that technique was too confusing...
then another approach would be to plug in a really really big x value like x=50 and see which function gives you the largest output.
facundo3141592

The exponential function with the largest base, which is e^(4x).

How to know which function increases faster?

There is a really simple rule, the rate at which a function increases is given by the differentiation of the function.

Particularly, an exponential function:

f(x) = e^x

has the property that:

f'(x) = e^x

So the differentiation gives the same function, which means that the exponential function is one of the functions that faster increases.

Here we have two exponential functions:

  • 3^x
  • e^(4x)

Which of these increases faster?

The one with the larger base.

The second one can be written as:

e^(4x) = (e^4)^x

Where e^4 = 54.6

So this function has a base a lot larger than 3^x, meaning that this is the function that goes to infinity faster.

If you want to learn more about exponential functions, you can read:

https://brainly.com/question/11464095