can you tell me why this natural base equation is sometimes true?

Question

e ^ 8x over e ^ 4 equals e^2x

I am sorry for... putting math equation like this but...
the answer says this equation is SOMETIMES TRUE
can anyone tell me why?? and explain how to solve this kind of problem?? like... state whether the equation is true or sometimes true or never...

thank you so much

Answer 1

When you have an equation with a variable in it (like "x"), and they say it's "sometimes true," what it means is: there are some values of "x" that you could plug in that would make the left side equal the right side; but if you plug in some different values of "x", the left side will not equal the right side.

To put it in other words, a "sometimes true" equation is one that is true for some values of "x", and false for other values.

The way to determine whether an equation is "sometimes true," "always true," or "never true," is to try to solve it with algebra and see what kind of answer you get for the variable.

If the algebra leads you to a nonsense answer like "2=7", that means there are no solutions (i.e. the equation is never true for any "x"). An example of such an equation is:
2x = 2(x + 5)

If the algebra leads you to something that has exactly the same thing on both sides (like "x=x" or "5=5"), that means the equation is always true (i.e., it is true no matter what value of "x" you plug in). An example of such an equation is:
4(x+2) = 3x + 8 + x

In the case of the equation you posed: It is true if you substitute x=2/3, but it is false if you substitute any other value for x. That means it is "sometimes true."

Answer 2

if you're familiar with rules involving exponents you can see

(e^(8x))/(e^4) = e^(8x-4)

and 8x-4 = 2x when 6x=4 or x=2/3

Answer 3

(e^8x)/(e^4) = e^(2x)
e^(8x-4) = e^(2x)

this is only equal when 8x-4 = 2x or x = 2/3

Note that the trick here is (e^8x)/(e^4) IS NOT e^2x !