How to raise an entire expression to functions, e.g logarithm

Question

I got very confused on how to raise an expression in parenthesis to some power which is not a number, but a function of a number.

E.g Can someone explain what this expression simplifies to or if there is any way to open parenthesis??? $$\Big (1 + \frac{\epsilon}{2\log t}\Big )^{\log t}$$

Thank you

Answer 1

Even a function is to be intended as a number, for each value of $t>0$ the expression is indeed defined.

You can also write it as

$$\Big (1 + \frac{\epsilon}{2\log t}\Big )^{\log t}=e^{\log t\cdot \log\Big (1 + \frac{\epsilon}{2\log t}\Big )}=t^{ \log\Big (1 + \frac{\epsilon}{2\log t}\Big )}$$

Answer 2

What you wrote doesn't simplify.

Answer 3

It can't be simplified...if you are perturbed by the log function, make a variable change, substitute $t=e^x$

Your expression becomes :

$$\Big (1 + \frac{\epsilon}{2\log t}\Big )^{\log t}=\Big (1 + \frac{\epsilon}{2x}\Big )^{x}$$