Vorde said:No, you can't just take the log of both. You aren't dealing with an equality you are comparing two equations. For specific values, your procedure of deriving both and comparing when one is greater works fine, but you have to derive the original equations.
In general though, exponential growth (blank^x) grows faster than x^blank (whatever that is called), and you can prove that with limits.
Gengar said:Or simply consider derivatives:
d/dx (e^x) = e^x and d/dx(x^e)=ex^(e-1)
These tell you how fast each function is increasing at each x, hence you can work out which function is increasing faster at each x.
e^x and x^e are both mathematical functions. e^x is the exponential function where e is a mathematical constant approximately equal to 2.71828 and x is the power or exponent. x^e is the power function where x is the base and e is the exponent.
Generally, e^x increases faster than x^e. This is because the value of e is larger than the value of x, so when raised to a power, e will result in a larger number than x^e.
Yes, there can be situations where x^e increases faster than e^x. This can happen when x is a large number, as the difference between e^x and x^e decreases as x increases.
You can determine which function increases faster by comparing the values of e^x and x^e for different values of x. If e^x is consistently larger than x^e, then e^x increases faster. If the values of x^e become larger than e^x at some point, then x^e increases faster.
Yes, there is a mathematical proof using calculus to show that e^x increases faster than x^e for all values of x. This can be done by comparing the derivatives of the two functions and showing that the derivative of e^x is always larger than the derivative of x^e.