Simple exponential function exercise

In summary, this conversation is about two equations that are equivalent. One equation is e^x^2=4, and the other equation is e^x=2. Both equations have the same solutions, which are x=ln2 for equation 1 and x=0 for equation 2.
  • #1
Adyssa
203
3

Homework Statement



is e^x^2 = 4 equivalent to e^x = 2

Homework Equations



As above

The Attempt at a Solution



This is just an exercise, but I'm quite stuck as how to show this is true (or false for that matter). I thought to take the log of both sides and use the log identity to get rid of the double exponent and cancel out the 'ln e' (=1):

ln e^x^2 = ln 4

x^2 ln e = ln 4

x^2 = ln 4

but I'm not sure this helps me (kinda went around in a circle!), nor am I clear on another method to use. I'm sure there's a log property I'm missing? :S
 
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  • #2
e^(x^2) and (e^x)^2 are two different things. e^x^2 doesn't mean much on its own. You probably mean (e^x)^2=4. Try taking ln of that.
 
  • #3
Oh my mistake, I should have been clear, it's e^(x^2) = 4 equiv to e^x = 2.
 
  • #4
I think you are asking does e(x2) = (ex)2

Let x=some appropriate number, say, 3, and use your calculator.

Re-examining, ex2 = (ex)2
you are really asking does x2 = 2x

Well, it does if x=2 or 0 :frown:
 
  • #5
I have to learn how to TEX >.<

The question is to show an equivalence between equation 1 and equation 2 where:

equation 1: e^(x^2) = 4 (so that's e to the x-squared equals 4)

equation 2: e^(x) = 2 (and this is e to the x equals 2)

So I think I have to manipulate equation 1 into the form of equation 2. To be honest, I haven't ever seen a question like this, we didn't do it in class, it's on an (unmarked) exercise quiz relating to basic algebra skills (which I blatantly lack, I keep gettng tripped up on stuff like this, just when I think I understand something!)

I should mention it's a TRUE or FALSE question, so they may not be equivalent, but I'd like to know! :P
 
  • #6
If the value of x that solves eqn 2 also solves eqn 1 (using your calculator), then there's a good bet that it is TRUE. Try that for starters.


Your restatement of the eqns amounts to what I wrote, in any case.
 
  • #7
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  • #8
OK so I did this:

e^x = 2
ln e^x = ln 2 (log law ln e^x = x ln e)
x ln e = ln 2 (ln e = 1, cancels out)
x = ln 2

and sub that into the first equation and sure enough I get 4, so they are equivalent. Now I feel like a right duffer! Thanks for the pointers though, I'm just getting a feel for logs and I get a bit flustered when I don't quite grasp the question.

:)
 
  • #9
Dick said:
e^(x^2) and (e^x)^2 are two different things. e^x^2 doesn't mean much on its own. You probably mean (e^x)^2=4. Try taking ln of that.
I disagree. (e^x)^2= e^(2x) which is an easier way to write it. I would immediately assume that e^(x^2) was meant.

And if (e^x)^2= 4 was meant, I would NOT take the logarithm. Since the "outer" function is squaring, I would take the square root first: e^x= +/- 2. Since an exponential (of a real number) cannot be negative, e^x= 2, x= ln(2) is the only (real) solution.
 
  • #10
Adyssa said:
I have to learn how to TEX >.<

The question is to show an equivalence between equation 1 and equation 2 where:

equation 1: e^(x^2) = 4 (so that's e to the x-squared equals 4)

equation 2: e^(x) = 2 (and this is e to the x equals 2)

So I think I have to manipulate equation 1 into the form of equation 2. To be honest, I haven't ever seen a question like this, we didn't do it in class, it's on an (unmarked) exercise quiz relating to basic algebra skills (which I blatantly lack, I keep gettng tripped up on stuff like this, just when I think I understand something!)

I should mention it's a TRUE or FALSE question, so they may not be equivalent, but I'd like to know! :P

No, they are not. The solutions of exp(x^2)=4 are x = +-sqrt(2*ln(2)) = +- 1.1774, while the solution of exp(x)=2 is x = ln(2) = 0.6931. It IS true that x^2 = 2x for some special values of x, but not for those values that solve either of your two equations.

RGV
 

FAQ: Simple exponential function exercise

What is a simple exponential function?

A simple exponential function is a mathematical function of the form f(x) = a^x, where a is a constant and x is the variable. The exponent x can be any real number, and the base a is typically a positive number. This type of function is commonly used to model growth or decay.

What is the difference between a simple exponential function and a compound exponential function?

The main difference between a simple exponential function and a compound exponential function is that in a simple exponential function, the variable appears in the exponent, while in a compound exponential function, the variable appears both in the base and the exponent. This makes compound exponential functions more complex and allows for more varied behavior.

How do you graph a simple exponential function?

To graph a simple exponential function, you can use a table of values or a graphing calculator. First, choose a few values for x and plug them into the function to find corresponding values for y. Plot these points on a coordinate plane and then connect them with a smooth curve. You may also want to include a few additional points to see the overall shape of the graph.

What is the domain and range of a simple exponential function?

The domain of a simple exponential function is all real numbers, meaning that any value can be plugged in for x. The range, however, depends on the value of the base a. If a > 1, the range will be all positive real numbers. If 0 < a < 1, the range will be all positive values less than 1. If a = 1, the range will be a single value of 1.

How can simple exponential functions be applied in real life?

Simple exponential functions can be used to model population growth, compound interest, and radioactive decay, among other things. They are also commonly used in statistics and probability to analyze data and make predictions. Examples of real-life applications include predicting the growth of a bacteria colony, the value of an investment, or the half-life of a radioactive isotope.

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